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THE 



MEASURE OF THE CIRCLE 



PERFECTED IN JANUARY, 1845, 



BY 

JOHN &A. 



PROVIDENCE: 

PUBLISHED FOR THE AUTHOR. 

1854. 



Entered, according to Act of Congress, in the year 1851, by 

SABIN SMITH, 

In the Clerk's Office of the District Court of the Southern District of New York. 

also , 

Copyrights secured for England, France, and Belgium, with the right of translation. 









THE MEASURE OE THE CIRCLE. 



THE USE AND IMPORTANCE OF THE MEASURE, 
DISCOVERED IN JANUARY, 1845. 

The globe is divided into 360 degrees ; each degree into 69^ 
miles; how can it be known what the 360th part is? A 
degree is a 360th part ; a mile is a 69^- part of a degree ; a 
yard is a 1760th part of a mile ; a foot is a third part of a 
yard ; an inch is a twelfth part of a foot ; a barleycorn is a third 
part of an inch. Now, these are parts which are all unknown, 
and have been since the world began, as may be supposed. 

I therefore say that all measures are imperfect, and, without 
a perfect quadrature of the circle, must remain so. I find the 
table of long measure is all imperfect, and so it is with all 
measurements, of all descriptions. There is nothing in shape 
to represent the noble works of Supremacy, known to man, 
that can be mathematically measured : such as land, if in a 
circular form ; casks containing liquids ; steamboat boilers ; 
a grindstone. These cannot, by mathematics, be correctly 
measured. 

Being encouraged to make this discovery by the large offers 
and appropriations by all governments, I devoted my best 
endeavors to effect a perfect quadrature ; and have, as I believe, 
to the satisfaction of all who may examine my work, and to 
my own, beyond all possible doubt. As I have learned, this 
measure has been the strife and anxiety of all the most learned 
men that have lived since the world began ; and, in defiance 

(3) 



4 THE MEASURE OF THE CIRCLE. 

of all, it has slept in oblivion until the year 1845, when discov- 
ered, as is represented in this work. 

I will set forth some of the objections that were made to me, 
to prove the impossibility of this measure, by different persons. 
A Mr. Clifford, the ninth wrangler of Cambridge University, 
said it was an utter impossibility, for it was a true figure of 
eternity ; and, as eternity never could be measured, no more 
could a circle. My reply was, " I admit it to be a true figure 
of eternity, as there is neither beginning nor end to it; but 
there is a great difference in measuring it, for I can get at either 
side of a circle, and neither side of eternity." The gentleman 
replied, ""Well, that does make a material difference." A Pro- 
fessor Olmsted said he thought it to be an impossibility, for it 
had been tried hundreds and thousands of years. My reply 
was, that it must be in nature, because I could get wrong upon 
the extreme, either way ; and there must be a point between 
those two extremes that must be right. 

And I say, in confidence, I have found the point, 

I was advised to introduce my work at Cambridge, in Eng- 
land. I did so ; and was told that the impossibility was so 
great that a man might as well try to shoot the moon. 

I was recommended to G. B. Any, professor royal at Green- 
wich. I called on this gentleman. He replied to me that it 
was not worth while for me to puzzle my brains about the 
measure of the circle, for it was measured as near as it ever 
could be ; and he did not see what use it was, or ever could be, 
to them. 

Now, it seems to me that this gentleman must be in an error, 
and grossly mistaken ; for I cannot see how he can pretend to 
hold out to the people a complete traverse of the stars, with- 
out the knowledge of the measure of the circle, unless he may 
think it as well to teach the world a falsehood as the truth. 
His situation in life ought to warrant to the people better abil- 
ity, and different expressions. 

I now come to some facts as to the use and principles. 

It is often said, in the construction of a steam engine, that 



THE MEASURE OF THE CIRCLE. 5 

the power is not equal to the calculation. They say the cause 
is, the steam does not properly condense ; and they allow fifteen 
pounds' weight of air to each and every square inch. Now, 
what can be expected, when they know not how many inches 
their cylinder contains ? 

My experience as a mechanic has taught me the use and 
importance of this measure. I have been engaged and employed 
as a superintendent, having the care and sole management of 
cotton, linen, woollen, and silk machinery, and have gained the 
approbation of my employers, my management being such as 
to save to them, yearly, a large amount. They said they could 
not comprehend how I could manage to such perfection ; for 
when I had perfected a thing, it was sure to answer the pur- 
pose. This perfection I arrived at by the measure of the circle, 
unknown to them. % 

The expression was made use of to one of my employers, in 
a cotton manufacturing village, " How is it that it does not cost 
you one half to keep your machinery in order that it does the 
rest of us ? " The answer was, " When Davis calculates, it is 
sure to come right ; while others have to do their work five or 
six times over." 

I was called on to measure a circular stair rail. The gentle- 
man wanted to know if I could calculate, by figures, the exact 
length of the rail. My reply was, that I could, if he gave 
me the height of the well. He gave me the height and diam- 
eter of his well. I gave him the length of the rail. He said 
I was wrong ; it was but so long. u Well," said I to him, 
" then your rail is that much too short." " It is," he said; " I 
have tried it." This I consider to be a demonstration by mathe- 
matics that cannot be performed except by the perfect quadra- 
ture of the circle. 

It is said that Sir Isaac Newton's opinion with respect to the 
mechanical powers was, that whatsoever was gained in time was 
lost in power ; and whatsoever was gained in power was lost in 
time. This maxim does not hold good in all cases. Sir Isaac 
was well aware that gravitation was not known ; neither can it 
1* 



6 • THE MEASURE OF THE CIRCLE. 

be, without the measure of the circle ; which when perfected, 
Sir Isaac's maxims are not correct, on account of the want of 
this measure — the long-sought solution of the perfect quadra- 
ture of the circle. 

"With these facts and circumstances, I shall leave to whoever 
may be interested in my welfare and interest to do for me what 
conscience may dictate to an honest heart. 1 ask or claim 
nothing but what good reasoning may, with sound judgment, 
honestly demand. I think I know the value and importance 
of this work, as my forefathers have, in all ages of the world. 
Utterance refuses expression, to paint in letters the utility and 
magnitude of this measure, which has been so long sought for. 

It appears, through and by the wisdom of God, that this 
circular principle is what he has put forth in his wonderful cre- 
ation of all things ; and why this measure has slept in oblivion, 
unknown to man, is known only to God. 

This measure will and must prove a great benefit to mankind, 
when understood, as it is the basis and foundation of mathe- 
matical operations ; for, without a perfect quadrature of the 
circle, measures, weights, &c, must still remain hidden and 
unrevealed facts, which are and will be of great importance to 
rising generations. The improvements that will arise from this 
measure fifty years hence I cannot paint in imagination. 

THE PRINCIPAL RULES. 

I think that, after people become satisfied that my work is 
right, it will be but one hour's labor for the scholar to learn all 
that is necessary in practice. I will, before I lay down the 
work in the book, lay down the principal rules for the learner, 
that he may see it in the commencement as well as in the last 
pages of my book. 

The use of the measure of the circle is to find the circum- 
ference of any circle, great or small, in order to correct and 
make right all weights and measurements, which arc wrong, 
and have been since the world began. 

The proportion the diameter has to the circumference is as 6 



THE MEASURE OF THE CIRCLE. 7 

to 19. The difference of as 7 to 22, or as 6 to 19, is as T th- is 
to TtrF* This makes linear measure 75f, square measure 1.52^, 
cubic measure 2.29 hundredths per cent, astray. This makes 
the foot rule near T V of an inch too short ; the yardstick near 
i% too short. 

A mathematical inch is the 38th part of a circle 12 inches 
in diameter. 

Rule 1st. To find the circumference of any circle, great or 
small. — Multiply the diameter by 9 T 5 ^ (this is my ratio/ de- 
rived from as 6 to 19,) and divide the product by 3 ; this gives 
you the perfect circumference, in all cases. Suppose your 
circle is 12 inches in diameter : — 

12 
9.5 

108 
6 

3)114 

38, circumference. 
Rule 2d. To find the area of the same circle. — Take 3 
times the radius, by once the radius; this gives the square 
inches. Divide the square inches by 3 raised to its 4th power, 
or biquadrate. Then add the 4th power, or biquadrate, to the 
square. This gives the perfect area, in all cases. Suppose the 
diameter is 12 inches, radius 6 : — 



6 
3 






18 
6 






3)108 






3)36 
3)12 




Add to 108 
4 


4, 


power. 


112 : 



8 THE MEASURE OF THE CIRCLE. 

Rule 3d. — Or you may take J of the square of the diam- 
eter ; that will give the perfect area, in all cases. Suppose the 
diameter is 12 inches : — 

12 
12 

144 

7 

9 ) 1008 



112, area. 
Rule 4th. Having the circumference, to find the diameter. 

— Divide the circumference by 19, and multiply the quotient 
by 6, which gives the diameter, in all cases. Suppose the cir- 
cumference is 38 : — 

19 ) 38 ( 2 
38 6 

12, diameter. 

Rule 5 th. — Let the cooper take 3 times the diameter, with 
one third the radius ; his head will just fill. 

Rule 6th. — Cut a thin strip of brass or tin, — the thinner 
the better, — 38 inches long, form a perfect circle, and the 
diameter is 12 inches. 

Rule 1th. Having the area, to find the circle that bounds it. 

— Suppose any number of figures as an area ; for instance, 
let it be 448 ; divide by 28 ; subtract the quotient from the 
given sum ; divide the remainder by 3 ; then extract the square 
root, which will be the radius of the circle that bounds the 
figures. The square root of any number of figures operated 
on in this way will be the radius of the circle, in all cases. 

144 ( 12, V 28 ) 448 ( 16 448 

1 28 16 



044 


168 


3)432 





168 





44 





144 


— 










Radius, 12 X 2 = 


24. 



THE MEASURE OF THE CIRCLE. 



MEASURE OF THE CIRCLE. 



Time is prefigurative of the number 6 ; so, as time is meas- 
ured by a circle, I take that number to measure a circle. 

Suppose a circle to be 12 inches in diameter. I take my 
radius, and multiply 12, diameter, by ratio 9-^ ; the product is 
114 ; divide this by 3, which gives 38, the circumference. 

Proof. — I begin with a hexagon, each side and radius being 
of equal length, say six inches. 6 multiplied by 6 is equal to 
36 ; this is the sum of the 6 sides of the hexagon. I call 
every polygon 36 inches, when it is 12 inches in diameter. 
I begin with a figure having 4 sides, it -being easy to under- 
stand. Take a strip of paper 36 inches long, and J of an inch 
wide ; cut it into strips of 9 inches each, and place them so as 
to form a square; the four corners will then be vacant. To 
supply these, I take £ of the radius, which is equal to one inch, 
and, for variety, call it the 7th. So I square the 7th by the 
number of sides. Multiply 4 by 4, — 16, square £. I add 1 
for each angle, to fill the vacant corners, which lengthens the 
square from 9 to 9^ inches. 9£ multiplied by 4 is 38, which 
is the circumference, when curved to a circle, gaining 2 inches. 
As every polygon will gain 2 inches, if worked right, I add 2 
to 36, which gives 38, circumference. 

I now begin with 6 sides and square the 7th, thus : 6 multi- 
plied by 6 is equal to 36 ; square sixth coned thirds. Add 1 
for each side, which is equal to 2 ; 2 added to 36 is equal to 
38, the circumference. 

I now take 8 sides, and square the 7th, thus : 8 X 8 ■=. 64, 
the 8th coned, \. Add £, equal side, =2. 2 added to 36 is 
38, the circumference. 

I take 10 sides, and square the 7th, thus : 10 X 10 — 100, 
tenth coned fifth, | for each side. 2 added to 36 equal 38, the 
circumference. 

I take 12 sides, and square the 7th, thus: 12 X 12 = 144, 
twelfth coned sixth, £ each side. 2 added to 36 equals 38, the 
circumference. 



10 THE MEASURE OF THE CIRCLE. 

This is to prove that the diameter is to the circumference as 
6 to 19. Now, I take 6 for my diameter, and suppose it to be 
formed into a hexagon. The 6 sides equal 18 inches, and 18 
multiplied by 18 is equal to 324. Add the square of the 
diameter, 36, to 324, = 360, the number of degrees in a circle. 
Now, the square root of 360 is equal to !8§f, or T V ; as there 
are no corners wanting to a circle, it leaves 19 inches for the 
measure of the circle. The circumference of the circle is 19 
inches ; but the square root of 360 lacks -gV ; so you see my 
diameter is 6 inches, and my circumference is 19, which makes 
it as 6 to 19. The measure of the circle is but linear measure ; 
therefore 36 makes an inch on a line, in this case, as is now 
used. Now, in extracting the square root of 360, it lacks 1 of 
filling the square which is superfluous in a circle, as no corners 
are wanting. 

I will now show how I came by my ratio. 

I take the 19, and divide it by 6, and when divided, it will 
come to 3 whole numbers, and 166§, decimal. Now, multiply 
3.166§ by 3. 3 times § is 2 whole numbers in decimals. I 
multiply by 3, in order to bring it to whole numbers, so as not 
to use fractions ; and, as I multiply this product by 3, I must, 
after multiplying the diameter by the ratio, divide by 3. 

12 
9.5 

108 
6 

3)114 

38, the circumference. 

THOMAS JEFFERSON. 

I find, in a work published by B. L. Raynor, in New York, 
in 1832, entitled " The Report of Thomas Jefferson to Con- 
gress, on Coins, Weights, and Measures, in 1790," that Mr. 
Jefferson says, in relation to all weights and measures, that " all 



THE MEASURE OF THE CIRCLE. 11 

are imperfect under the present system." The report from the 
secretary of state, containing a plan for a uniform system of 
coins, weights, and measures, on page 311 of this book, was 
executed with most astonishing despatch, considering the intri- 
cacy of the subject, and novelty of the experiment. In sketch- 
ing the principles of his system, Mr. Jefferson was dependent 
on the guide of his own genius, as no example to dictate or 
direct his researches existed. It is somewhat remarkable that 
two of the principal governments of Europe were also engaged 
at this period on the same subject. 

The first object that presented itself to his inquiries was the 
discovery of some measure of invariable length, as a standard. 
There exists not in nature, as far as has been hitherto observed, 
a single object, accessible to man, that presents one uniform 
dimension. 

The globe of the earth might be considered as invariable in 
all its dimensions, and that its circumference would furnish an 
invariable measure ; but no one of its circles, great or small, is 
accessible to admeasurement, in all its parts ; and the various 
trials to measure different portions of them have resulted in 
showing that no dependence can be placed on such operations, 
for a certainty. Matter, then, by its mere extension, furnishes 
nothing invariable. Its motion is the only remaining resource. 

The motion of the earth on its axis, though not absolutely^ 
uniform and invariable, may be considered as such, for all 
human purposes. It is measured, obviously, but unequally, by 
its departure from a given meridian of the sun, and its return 
to that meridian, constituting a solar day. Throwing together 
the inequalities of solar days, a mean interval, or solar day, has 
been found, and divided by general consent into 86,400 parts, 
called seconds of time. 

Such a pendulum, then, becomes itself a measure, of deter- 
mined length, to which all others may be referred, as a standard. 
But even the pendulum was not without its uncertainty, as the 
period of its vibration varied in different climates or latitudes. 
To obviate this objection, he proposed the standard might refer 



12 THE MEASURE OF THE CIRCLE. 

to a particular latitude ; aud that of 38 degrees being the mean 
latitude of the United States, he adopted it. 

THE CIRCLE. 

The circle is one of the noblest representations of Deity, in 
his noble works of human nature. It bounds, determines, 
governs, and dictates space, bounds latitude and longitude, 
refers to the sun, moon, and all the planets, in direction, brings 
to the mind thoughts of eternity, and concentrates the mind to 
imagine for itself the distance and space it comprehends. It 
rectifies all boundaries ; it is the key to information of the 
knowledge of God ; it points to each and every part of God's 
noble work ; it divides east from west, north from south, with 
all its variations so beautiful ; it brings to the thoughts of man 
the eternity and incomprehensibility of that space and distance 
which none can explore or determine. Nothing but imagina-' 
tion can cope with its extent, bounded as it is by the most ex- 
treme, unseen and unsought distance that minds can imagine. 
It has neither beginning nor end ; its bounds are unknown ; its 
area cannot be told by numbers ; no mouth can reveal its mag- 
nitude ; it is what contains all human flesh and blood ; it con- 
tains all the improvements susceptible to the ingenuity, science, 
and activity of the human family ; it brings to mind that eter- 
nity of bliss and happiness where the weary are at rest, and 
from whence no traveller returns ; it contains monuments of 
marble and stone, in memory of those who have sought its 
quadrature in all ages of this world ; it is the companion of 
every man, woman, and child ; it enters the families of all the 
earth ; it is the mediator of honesty, harmony, and content, in 
all ; it rectifies those principles which are calculated to comfort 
and console honesty to the bosom of all. Its want of correct- 
ness has been the strife and anxiety of all the learned, and the 
lovers of science, since the world began. The wrongs which 
man has done to man, for want of a perfect measure, are 
numerous. Immense sums of money, with much time and 
anxiety, have been spent by all nations for its perfection ; and 



THE MEASURE OF THE CIRCLE. 



13 



no one nation, perhaps, is worthy of more respect for its exer- 
tions than France. 

Its proportion is now found. It can be measured to perfec- 
tion. It can no longer slumber, for its equality can now be 
expressed. 

DIAGRAMS OF CIRCLE. 




This is a hexagon, having 6 equal sides. 



llllilllllllllllllll!l(llllllllllllllllHlllllllillll!|||||||i | |liliillHllillillllllllllll 



i^iiiiii.'iiii'iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiji EH 



This is to describe the four strips of 
paper set forth in the work, supposed to 
be 9 inches square, leaving the vacant 
corners as described. 



The hexagon is what I measure the circle by, it being the 
only figure known by which the circle can be measured, and 
the number 6 the only number. 



This is an inch divided 
into sixths. 




This is an oblong square, 6 by 3 



14 



THE MEASURE OF THE CIRCLE. 



18 
6 

3)108 

3)36 

3)12 



This is the 4th power, or biquadrate. Add this to the square, 
and it gives the area of the circle. 

12, diameter. 108 
4 

112 
The question has been asked by many, why some other fig- 
ure will not answer as well as the hexagon. The hexagon is 
equal in all its parts, and no other figure is. Had I taken any 
other, it would have carried me into surds, which would be 
beyond comprehension. 




An equilateral triangle, 
having 3 equal sides. 





THE MEASURE OF THE CIRCLE. 



15 



This is a rectangle, having 4 
sides, and 4 right angles. 



An acute angle, sharper than 
right angle. 






An obtuse angled triangle, having 
one obtuse angle. 



A rectangle, having 
4 equal sides. 



16 THE MEASURE OF THE CIRCLE. 



TO MEASURE A CIRCLE BY RINGS. 

Suppose the diagram is a triangle, £ of an inch scale. To 
measure this angle, I divide the circle into 36 rings, each ring 
{ of an inch wide, and take the measure of the 18 th ring, 
which is as follows : Take 112, which is the area of the circle, 
12 inches being the diameter. Now, to find this area, I take 
the square of the radius, which is 6 inches, and multiply it by 
6, = 36 ; multiply this 36 by 3, = 108 ; divide this 108 by 
3, = 36 ; divide this 36 by 3, = 12, which is the cube, or 
third power ; divide this 12 by 3, = 4, which is the biquad- 
rate; then I add the biquadrate to the square, 108, and it 
gives, the perfect area, thus : 108 + 4 = 112, area of circle. 

Now I multiply this area, 112, by 6, which throws the area 
into a strip 672 inches long, and i of an inch wide ; this I 
divide by 36, the number of rings in the circle ; this gives the 
average length of each ring, on a straight line, which is 18| 
inches, but on a curve it is 19 inches. Now, this 18th ring, 
is, when lengthened out, 18| inches, but when brought to a 
circle, 19 inches. The reason for this is, that when you change 
this straight line to a circle, you take, in order to make the 
concave, f of an inch from the inside, which goes to the length 
of the ring, to make it 19 inches. Now I will show the length 
of the first and second ring. Take the diameter of the first 
ring, which is 36, multiply it by 9.5, =s 342 ; divide this by 
3, z= 114, length of the first ring. For the second ring I take 
35, and multiply it by 9.5, = 110.833^. Now, in order to 
make the work shorter, I take the 18th ring, as being the 
average length of the 36 rings, which I have shown above, 
and multiply its measure, 19 inches, by the radius, 6 inches, 
thus : 19 X 6 = 114 inches, area of the angle, but not of the 
circle, as I shall prove. The circumference of the outside 
ring is 38 inches; multiply this by 3, which gives 114. Sec- 
ond ring, 110.833^, which is just 3£ less, equal to 1 T V on a 
circle, but only 1 on a straight line. So the triangle measures 
T V, which is equal to ^ too long. I therefore take twice 36, 



THE MEASURE OF THE CIRCLE. 



17 



which is equal to 2 inches, from 114, which leaves 112, area 
of circle. 

This is a diagram one sixth of an inch scale. 




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Ci CO C5 CO C» CO G5 CO Ci CO C5 CO 05 CO Ci CO OS CO Ci CO Ci CO Oi CO 
Oi CO Ci CO Oi CO Ci CO Ci CO Ci CO Ci CO Ci CO Ci CO Oi CO Oi CO QM 
WKO-*. W|M&». KIN**. WfWW^- KJ<«|f- C*«CO>- eqtttqh- W|NM«- MjNW»- U|NW>- WRite^- «|tC«H 



SIT a; 



8?I 



Each and every one of these rings loses in length one inch 
and T \. Every ring is one inch and T V shorter inside than on 
the outside. 

Time being prefigurative of the number 6, I take that num- 
ber to measure the circle. Suppose my radius is 6 inches, by 




18 THE MEASURE OF THE CIRCLE. 

£ of an inch ; then my circle will be 12 inches, which, brought 
to a hexagon, will measure 36 inches, equal to the square of 
the radius. Now, I divide one side of the hexagon into 36 
squares, and, supposing my radius fixed by one corner to a 
point at the centre of the circle, I move the other end just its 
width, which is £ of an inch on a straight line, which gives the 
biquadrate, equal to ^V of the radius ; and, as it doubles on a 
circle, it is equal to T V gained on the side of the hexagon, equal 
to s 2 F , which added to 36 degrees, make 38, the circumference, 
or 6£ inches for £ of the circle. 

So 6 prefigures time ; time, with its square, with the biquad- 
rate conically added, measures the circle ; therefore, to 3 times 
the diameter, equal 36, add the square root of the biquadrate, 
equal 2, which gives the circumference. Also, to 3 times the 
square of the radius, 108, add the biquadrate, 4, which gives 
112, the perfect area of the circle. 

OBSERVATIONS ON THE MEASURE OF THE CIRCLE BY RINGS. 

1. Every ring £ of an inch wide loses £ of an inch, upon an 
average, when straightened out, by reason of the inside being 
shorter than the outside. Suppose a ring be 19 inches long; 
18 inches and f would be the measure of it on a straight strip, 
square at the ends. Every ring being ^ of an inch wide, and 
fitting one within another, being 36 in number, they must be 
just 12 inches in diameter, and 38 in circumference ; and the 
average measure of each ring being 3 and ^, and 36 the num- 
ber of rings, the area must be 112. If you wish to measure a 
ring separately, take the measure of a straight strip £ of an 
inch wide, and subtract -£ T part of the circumference from it, 
and the remainder will be the area of the ring. 

2. Every circle bears the same proportion to another as a 
square. As, suppose a square 6 inches, and another 12 inches ; 
the square of the 6 inch would be 36, and of the 12, 144, or 
4 times as much ; and if it was 24 it would be 16 times as 
much. So, in a circle 6 inches in diameter the area is 28 
inches; but 12 is 4 times 28, equal to 112; and 24 is 16 



THE MEASURE OF THE CIRCLE. 19 

times as much, or 448, the area, although the circle would 
only double, as, 6 diameter, 19 circumference, and 12 diameter, 
38, &c. A line to go round a circle would be just in the same 
proportion. 

3. To find the measure of the circle. Suppose 12 is the 
diameter ; then the radius will be 6. Now find the biquad- 
rate, thus : — 



36 
6 

216 
6 



1296, the biquadrate, or square of 
the square, as 36 X S6 = 1296. 

Now take a hexagon, divide one side into 36 sixths of an 
inch, and the radius | of an inch wide. Cut one side so that 
you can move it its width to its centre point. The outside 
angle being on the hexagon, by moving the centre % to the 
centre point, you raise £ from the next angle ; but as it must 
meet the angle again it gains double the biquadrate, just the 
same as the square ; so that instead of gaining ^V* ** gains T \ 
of the hexagon, or any part of it. And this is the perfect 
measure ; for as it proceeds from its own centre, there cannot 
be any defect in the measure, any more than in the circle 
itself; for at the same time that it gains ^ on a straight line, 
it gains ^V tne other way, which when curved will be just T \ ; 
and 36 sixths on a straight line, and twice on a curve, equal 
38 sixths, one side of the hexagon, just 38 inches the whole 
circle. 

4. The square of a circle 12 inches in diameter is 9£ inches ; 
this is the square of the circle, but is not the square of the 
area of the same circle. The square of the area of a circle 12 
inches in diameter is about 10.583 inches. 



£0 THE MEASURE OF THE CIRCLE. 

The fourth part of a circle is the square of the circle itself, 
but not the area. The circle gains over all other measures. 
Take a circle 38 inches round, the diameter of which is 12 
inches, and the area of this circle is 112 inches. Now, take a 
square 38 inches round, that is, 9£ inches on each of its four 
sides, and the area will be 90^ inches. The area of a circle 
12 inches in diameter will be 112 inches; and of a circle 6 
inches in diameter, 28 inches. So you see that it varies the 
same as the square. Half the diameter is J in area ; also, 3 
times the diameter gains the biquadrate in circumference, and 
also its square in area ; as, 2 X 2 = 4, area. 

Now, this is strong proof of the perfect measure of the cir- 
cle, and beyond a doubt ; because as the square varies, so varies 
the area ; they are synonymous ; one proves the other, in all 
cases. 

5. From Euclid. The impossibility of expressing the exact 
proportion of the diameter of a circle to its circumference by 
any received way of notation has put the most celebrated men 
in all ages upon approximating the truth as near as possible, 
there being a necessity of a nearer quadrature, inasmuch as it 
is the basis upon which the most useful branches of mathe- 
matics are built. 

The famous Von Keulen carried it to 36 places of decimals, 
which he ordered to be placed on his tombstone, thinking he 
had set bounds to further improvements. The next that 
attempted it with success was the indefatigable Mr. Abraham 
Sharp, who, by a double computation, viz., from the sine of 6 
degrees one way, and from the sine and cosine of 12 degrees 
another way, carried it to twice the number of places that Yon 
Keulen had done, viz., 72. Afterwards, Professor Machin, by 
different methods of computation, carried it to 100 places. 

Now, it is spoken of as a most wonderful feat of exactness, 
of scientific mathematical knowledge, to perfect a complete 
measure of the circle ; and I have heard learned men say, " I 
have carried it to 50 and 100 places." Archimedes did not 
suppose that he had found a perfect ratio or an exact propor- 



THE MEASURE OF THE CIRCLE. 21 

tion of the diameter to the circumference of a circle, when he 
said it was nearly as 3 times, or 7 to 22. 

What probability is there to a mathematical mind that by 
running the figures out — if it were possible — it would effect 
a perfect measure of the circle, as long as the exact proportion 
of the diameter to the circumference was not found ? 

In order to make a ratio that will measure a circle correctly, 
you must first have the exact proportion the diameter has to the 
circumference, as the ratio is for a common multiplier, to find 
the circumference, and is to be derived from the exact propor- 
tion the diameter has to the circumference of a circle. I have 
found, by the operation of figures, that this proportion is as 6 
to 19. Now, in order to make a ratio, I divide the 19 by 6, 
which gives 3.166£. I make a vulgar fraction of -£, and reduce 
it to §. I find the value of 3.166| by multiplying by 3, which 
brings it into whole numbers, 9£. This as a ratio will answer 
for whole numbers in all cases, because it finds the substance of 
both dividend and divisor. My proportion of the diameter to 
the circumference is perfect, and that being perfect, makes my 
ratio perfect. But any ratio to be derived from as 7 to 22 
would be imperfect, inasmuch as the proportion is imperfect. 
The ratio to find the area of a circle is 7854. Reason must 
dictate to all that square measure is not circular, because an 
angle of 90 degrees is square measure, and an angle of 60 
degrees is circular. 

To perfect the exact area of any circle by square measure, 
which is what this ratio is derived from, is impossible, from the 
nature of things, as by this ratio it makes the area of a circle 
12 inches in diameter to be 113 T V inches. This is just square 
measure ; and circular measure of the same circle is 112 mathe- 
matical inches. Now, you see these angles differ 30 degrees. 
See the following demonstration : 12 being the diameter, they 
make the circumference 37 T 7 ^, and the area 113 T V- Suppose 
them divided into 36 rings ; divide the circumference thus : 
37.7 t2= 18.85, mean length of rings. Now, multiply by 
the radius, 6 inches, which will give 113 T V inches, area of the 
rings, which is exact square measure, instead of circular. 



22 THE MEASURE OF THE CIRCLE. 

By mathematical attention to these facts you will see that no 
square measure can perfect circular, which is what measures the 
circle ; and you will find it impossible to perfect the measure 
of the circle by square measure. 

In a circle 12 inches in diameter form a hexagon and a 
dodecagon, and take one of these angles, and the angle of the 
hexagon and dodecagon will measure just 18 inches the 6 
angles ; the 6 angles added together will be 108 inches. Now, 
the segment of this circle will measure just 4 inches, which is 
the 4th power, or biquadrate, and makes up the measure of the 
circle. That is its area ; 3 times the diameter, with the biquad- 
rate added, is the perfect circumference. Also, 3 times the 
square of the radius, with the biquadrate added, is the perfect 
area. 

Dividing by 3 to the 4th power, is the same as dividing by 27. 
It is the j/f part added to the square ; as 3 X 3 — 9, and 9 X 
3 = 27. The biquadrate is T \ of 3 times the diameter of the 
circle ; and the biquadrate is also 2 y part of 3 times the square 
of the radius, which makes up the area. 

6. To convince that 90 degrees is a wrong angle to measure 
a circle, and to find the true area of a circle 12 inches in diam- 
eter. The largest ring in the plate of the 36 rings is supposed 
to be 38 inches in circumference, and the 18th half the size, or 
19 inches. Now, the way they measure is to multiply half the 
circumference, equal 19, by the radius: thus, 19 x 6^:114. 
By this you measure every ring by square measure, that is, 90 
degrees. Thus, 

19 inches long, and 1-6 wide. 
| | equal 3 1-6 inches. 

Multiply 36 rings, the number of rings in the circle : — 

36 
H 

108 
6 

114 inches. 



THE MEASURE OF THE CIRCLE. %0 

But a circle falls short of the measure £>f a straight strip, as the 
inside is shorter than the outside. So proceed thus : take the 
hexagon 6 inches, as laid down in the work ; the circle is 19 
inches, 

19 long, and 1-6 wide. 
f -J 

Set a bevel to 60 degrees, and place it on the outside of the 
hexagon, and you will find it falls short of 90 by 30, equal to 
£ to each angle ; so the 6 angles lose -g$ of an inch ; and 19 
inches being the average length of the rings, each ring loses ^ 
of an inch, which multiplied by 36 are equal to 2 inches, to 
be subtracted from 114, leaving 112, the perfect area of the 
circle. 

This is the measure of the area of a circle 12 inches in diam- 
eter ; and if the area is right, the circumference by which the 
area is bounded must be right also, for one proves the other. 

7. To show the error of the old trigonometrical way of meas- 
uring the circle : Take 12 for diameter ; suppose it to be made 
into 36 rings, as before shown ; then the longest ring is 38 inches 
on a line, on the outside, and every ring is T *g shorter on the 
inside than on the outside ; so the 36 rings lose just -f -| of an 
inch, equal to 2 inches from 114 — leaving 112 for true area; 
but in their way the 12 inch ring, instead of 38 circumference, 
is but 37/q. Now, divide 37 T 7 - by 2, equal 18.85, the mean 
length of rings. Divide them into 36 straight strips 18.85 long, 
i of an inch wide, square at the ends, equal to 6 inches wide ; so 
18.85 multiplied by 6, equals 113^, area; but if made into 36 
rings, their area would be but lll-jj^: 

8. To prove that the present measure in use, that makes the 
circumference of a circle 12 inches in diameter to be 37 T \, 
is perfect square measure, because it makes the area of this 
circle 113^ inches, I draw a diagram, and suppose it to be 
37 -/„ inches long, and 6 inches radius. I make two angles, 
which will be 6 inches at one end, and a point at the other, and 
one of these angles will be just half the oblong square, which is 
113^ inches, according to the measure in use. 



u 



THE MEASURE OF THE CIRCLE. 



Circumference, 37 7-10. 




37.7, circumference. 
6, radii. 



2 ) 226.2 



113.1, square measure. 

To show the common error of the measure now in use, which 
makes 113 T V the area of a circle whose diameter is 12, and cir- 
cumference 37 1 7 <y, which area is 1 T V = •}£ too much, I will per- 
form an operation by tenths. 



)110(28ff 


28 


36 


76 


2 


2 


340 


56 


2)72 


304 


1 


— 





— 


38 


36 


WH 


34 



This shows that if the circumference is 37^, and the area 
113 T V, the diameter should be nearly 12.057f|, instead of 12. 



A RULE FOR GAUGING. 

Having effected the measure of the circle, and knowing that 
circular measure has never been in use, and believing that the 
gauging of casks must be imperfect, many complaints being 
made of the present gauging, I have investigated the subject, 
and found the true and mathematical measure by the circle. 

By specific gravity, 62£ pounds of distilled water are 1 cubic 
foot ; and the law provides that a gallon shall weigh 8 pounds 
of distilled water, weighed at the level of the sea. I find that 
221 inches weigh 8 pounds, according to the statute gallon ; I 
also find the circular inches in a gallon to be 284. Now, in 



THE MEASURE OF THE CIRCLE. 25 

order to effect this measure, I procured a box nearly 6 inches 
square, and 6J- inches in depth; then I boiled some Croton 
water, and strained it through a flannel cloth ; I then poured 
into the vessel until it weighed 8 pounds, and found it filled 
the vessel 6§ inches, as near as I could determine by the eye, 
which I consider as perfect measure as could be obtained from 
such measuring. 

By mathematics I find that a vessel 6 inches square, in order 
to hold a gallon, must be 6 inches and £, and ^ part, in depth ; 
and I consider the T V part as not perceptible to the eye. I 
therefore come to the conclusion that the gallon is precisely 221 
inches, as I 'know it to be by mathematics, thus : — 

6 
6 

36 
6 

216 inches, lacking 5 of a gallon. 
5 I add the 5 inches. 

221, the square inches in a gallon. 

Now, i of 36 is 4 J ; and T V of 36 = £ = 5, which added to 
216 = 221 inches, or 8 pounds. 

To find the circular inches in a gallon. The circle is £■ of 
the square ; therefore 221 is £ of 284, which is f, thus : — 

7)221 
31f 



221 

284, the number of circular inches 
in a gallon. 



26 



THE MEASURE OF THE CIRCLE. 



Rule for measuring the contents of a cask. — Take the 
diameter of the head and bung, and add them together ; take 
£ of their sum, which will give the mean diameter. Multiply 
the mean diameter by itself, and that product by the length of 
the cask ; then divide the last product by 284, and the quotient 
will be the answer, in gallons and parts. 

For example: Multiply 31.5 by 28.7 = 3261636. Divide 
this by 284 == 114Jt£ gallons. 

The more to convince, I have obtained from one of the 
gauging masters fourteen casks, and compared them with my 
measure, and find the gauging to be Horn 1 to 5 per cent, more 
than my measure, which is circular. I have tried my measure 
by actual experiments, by measuring water into casks, and 
found it to be as perfect as can possibly be expected from such 
measuring. 



No. of cask. 
1 


Length. 


Bung. 


Head. 


Gallons. 


True measure. 


Difference. 


36.9 


32.3 


29.3 


121 


117 


1 


2 


35.7 


33.5 


30.0 


126 


126 





3 


37.3 


29.3 


26.0 


102 


100 


2 


4 


36.9 


31.5 


28.3 


117 


116 


1 


5 


35.0 


33.9 


29.4 


124 


123 


1 


6 


37.6 


33.4 


29.2 


129 


130 


1 


7 


37.9 


29.9 


26.5 


107 


106 


1 


8 


49.9 


31.6 


28.5 


130 


130 





9 


37.8 


33.3 


30.0 


134 


133 


1 


10 


38.8 


33.7 


29.2 


135 


131 


4 


11 


35.4 


33.3 


29.6 


125 


123 


2 


12 


40.0 


32.0 


24.0 


115 


110 


5 


13 


35.1 


32.3 


29.5 


118 


118 





14 


36.8 


32.7 


28.8 


124 


123 


1 



The above table represents the gauging of 14 casks, by 
sworn gaugers in New York city, as handed to me. I find, in 
this number of casks, that the gaugers make 20 gallons more 
than I do by my rule of gauging ; so that I think it will vary 
from 1 to 5 per cent, too much. 

To show the measure of a circle by rings. Suppose 12 is 
the diameter ; 12 X 9.5 =i 114 — 3 = 38, the circumference. 
Now, the longest ring is 38 inches long, and £ of an inch wide. 



THE MEASURE OF THE CIRCLE. 27 

Suppose 36 straight strips £ of an inch wide and 38 inches 
long would measure just the same as a piece of board 38 inches 
long and 6 inches wide. 



2)228 

114 

Dividing 228 by 2 would give 114, if the rings were all straight 
strips. But these strips are the same measure on both sides, 
while the inside of the rings is 3V shorter, reducing the meas- 
ure of the strips ■£-, equal to 2 inches ; therefore subtract 2 
from 114, leaving 112 for the rings. 

To prove my measure of the circle by the square, draw a 
square just 6 inches in diameter ; then 6 multiplied by 6 will be 
36, the area ; then draw another of 12 inches, which is 2 diam- 
eter ; but 12 multiplied by 12 is 144, just 4 times the area. 
So twice the diameter gives 4 times the area ; and whatever is 
the result in a square will be the same in proportion in a circle, 
if it is measured right. 

So you may see in my work that 6 for a diameter gives 28 
for the area, which is just 4 times the same, in proportion, as 
the square, which proves the measure correct, beyond a doubt ; 
for if the area is correct, it is impossible for the circle that 
bounds it to be otherwise. 

I have been so much troubled in trying to convince people 
of the measure of the circle, and its area, that I am almost 
prone to think that if some college-bred man were to say twice 
2 was equal to 5, many of them would not appeal to common 
sense, but conclude at once that it was so. For instance, a 
rectangle of 90 degrees is commonly called a right angle ; and 
I allow it to be a right angle for a square, but not for a circle. 
Its result would be 33^ per cent, astray. 

The right angle for a circle is 60 degrees. But they tell me 
that 60 is not a right angle ; and I know that 90 degrees is not 



28 THE MEASURE OF THE CIRCLE. 

a right angle for a circle. It would seem that they wish me 
to act the monkey or mimic, so that if they are wrong, I 
must be. 

Now to the point. A circle 12 inches in diameter is 38 
inches in circumference, as is proved in my work. By trigo- 
nometry, which is in the proportion of 90 degrees, I take the 
circumference for one side of the triangle ; the extreme width 
is 6 inches, and the mean width is 3 inches ; so I multiply 38 
by 3, which gives 114 for the area. 

I now come to circular measure. Take 6 times the measure 
of the oblong square, as laid down in this work, or 3 times the 
square of the radius, and either of them is just the measure of 
the dodecagon, or polygon of 12 sides, equal to 108, square 
measure. 

Now to measure the circular part. The 6 sides of the hex- 
agon measure 36 inches, and the circumference is 38, gaining 
£ inches, which is y F . I will work it, to show the difference : 

If 90 : T V : : 60 If T V : 6 : : & 

90 18 



60 ) 1620 ( sV, answer. 27 ) 108 ( 4, answer. 

120 108 

420 

420 

4 added to 108 equals 112, area of 

circle. 

Now, 112 is circular measure, and 114 is square measure, and 
in the proportion of 90 degrees; so you may divide 112 by 
36, which will give the mean area of each ring, or you may 
measure them separately ; and if the rings are measured right 
it must be the true measure of the circle ; and its area for the 
one will prove the other. 

The following diagram represents T V of a circle ; so I cut off 
the circular part, and divide the remainder into 3 parts, equal 
in length to the chord of the circle. Set your square on the 



THE MEASURE OF THE CIRCLE. 



29 



chord, and move the other end to the centre point, and it will 
leave an angle on the chord equal to T V 




THE CIRCLE. 

A circle is a line continued equidistant from a common centre 
till it ends where it began. The diameter is the line drawn 
from one side of the circle, through its centre, to the opposite 
side. The radius is the length of a line drawn from the cen- 
tre to the circumference. 

I think the most easy way for people to understand my 
measure of the circle is by the rings, which I have laid down 
in this work ; of the correctness of which I think I could con- 
vince any man that would listen to common sense. 

To find the area of a circle, divide the circle into six angles, 
or points, by the 6 radii, and take an oblong square from one 
of them, as before shown. Multiply that square by 6, and the 
product will be just the area of the dodecagon, or polygon of 
12 sides. Find the biquadrate by dividing the area by 3 to its 
4th power, and add the quotient to the area, and it will be the 
area of the circle, or space contained within the circle. 

With regard to measuring a circle by polygons, with the 
help of mathematics, it has been disputed, by saying the poly- 
gons varied, and that I have called them 36 inches, which I 
allow to be so ; for if one side of the hexagon is 6 inches, with 
6 sides, the side of a polygon which has 12 sides would be 3 
inches, to be in proportion. But it is S^i^M?' This dispute 
did not arise from ignorance, as I think reason must dictate 
that I should prefer rational numbers to work with, when they 
answer the best purpose. Now, we know that the square and 
3* 



30 THE MEASURE OF THE CIRCLE. 

the hexagon are both perfect figures ; and if the result of these 
is right, the result of all the others must be so, when worked 
the same way. And that they are so may be proved by the 
work ; as, whatever polygon you try, it gives the same measure 
of the circle ; and if it did not it would prove itself wrong. 

This dispute seems ,as unreasonable to me as it would for a 
man to dispute that twice 10 made 20, because they are not 
counted one by one. 

To find the circumference of a circle 6 inches in diameter : 
The 6 radii are 3 inches ; so I take the square : — 

18 
18 

144 

18 

324, square of 18. 
36, the square of the diameter. 

360 ( 18ff 
1 

28 ) 260 
224 

36 

So it wants -^ of s\ of being 19 inches. But 1 inch on a 
straight line is equal to 1 T V on a circle. So, when the T V is 
made to cover the two sides, which is always the case in the 
square root, it will be but -fa, which leaves the circumference 
just 19 inches. 




The area is 28.5 by trigonometry, as is used in the common 
way ; but circular measure makes it only 28. 



THE MEASURE OF THE CIRCLE. 31 



THE DIFFERENT RATIOS. 

To show the measure of a circle 12 inches in diameter : 
Suppose you draw a circle 12 inches in diameter with a pair of 
compasses, from which you draw a hexagon, one side of which 
will be 6 inches on a straight line, but on curving it to a circle 
it will be found, by dividing it into 36 degrees, instead of 60, 
to bring the fractions perfect, which fractions are equal, to 
double the biquadrate found. 

Thus the radius, 6 inches, 6 X 6 = 36 sixths of an inch, 
and one degree is £ of an inch, which added to 36 = 37, the 
hypotenuse, ^V of ^ on a straight line ; therefore in one side 
of the hexagon I gain just one degree, equal i of an inch, and 
on a circle two degrees, equal £ of an inch, as the circle always 
gains double the same, in proportion, as the square. Although 
the circle does not make the square abruptly, it does make it, 
or has to make it ; so the measure is mathematically true. 

Diameter, 12 inches; 6 times 38 degrees, equal J each — 
38 X 6z=228-^-6 = 38 inches, circumference. 

I will now measure a circle 12 inches in diameter by the 
measure now in use : — 

3.1416 
12 



37.6992 
This makes it 37 inches, T 6 -, T § 7 , and i^uu- 



I 

use: 


now 


find the 


area 


of the above 

7854 
144 


by 


the 


measure 


now 


in 




31416 
31416 

7854 





13.0976 
This is 113 inches, T ° ff , ythy, and tJut- 



Z2 THE MEASURE OF THE CIRCLE. 

I now find the circumference by my ratio, 12 being the 
diameter : — 

12 
9.5 

108 
6 

3)114 

38, circumference. 

I now find the area of the same ; diameter 12 inches, radius 
6 inches : — 

6 



18 
6 

3)108 

3)36 

3)12 

4, 4th power, or biquadrate. 
Add the 4th power to the square, 108 + 4 = 112, area of 
circle. 

The difference of ratio 3.1416 and ratio 9.5 is: Ratio 9.5, 
38 inches; ratio 3.1416, 37 inches, T 6 ^, T §^, and T uVu; differ- 
ence, 1.0976 inches ; difference of area or circumference of 
same, T 3 <y of an inch. 

So you see the measure now in use gives less circumference 
and more area, which is wrong, beyond all doubt. 

ARCHIMEDES' MEASURE. 

The proper ratio to be derived from as 7 to 22 is 3.1428f, 
which measures a circle nearer than any measure that has been 
in use since Archimedes left it. All the attempted approxima- 



THE MEASURE OF THE CIRCLE. 33 

tions have been farther off from the correct measure, instead, 
as has been thought, of being nearer. It has been a general 
prevailing idea that Archimedes' measure, as 7 to 22, was too 
much ; and all have been striving to make it less. 

I now find the area by rings, ^ of an inch wide. If \ou 
begin at the centre point, and lay ring upon ring, you may not 
perceive it; but if you begin at the outside you will find 112 
inches just meet the centre, which proves it right ; whereas, 
113.1 will run 6 T 6 g- inches beyond the centre, which proves it 
wrong, in spite of every thing that can be brought in its favor. 
Now, in a circle 12 inches in diameter, they make the circum- 
ference 37.7, and the area 113.1 ; I make the circumference 
38 inches, and the area 112. So they make 1 T V inch more 
area, and T 3 ^ less circumference. 

I should like to know what to do with the lyV inches in the 
area of a circle 12 inches in diameter, and call it 38 inches in 
circumference, much more call it 37 T V 

People seem determined to believe that the circumference is 
37.7 ; but all can see by this that the rings would of course be 
5.4 inches shorter, and would not reach the centre point by the 
5.4 inches. But according to the area 113.1, they would run 
6| beyond the centre, being that much too long ; but by short- 
ening the circumference and increasing the area, as has been 
done, the rings would run 12 inches beyond the centre point 
of the circle. But if the circumference and area were right, 
the rings would just strike the centre. 

I find the square root of a circle 12 inches in diameter : — 

112 (10.58^ 
1 

205 ) 01200 
1025 



2108 ) 17500 
16864 



636 The answer is 10.58^. 



34 THE MEASURE OF THE CIRCLE. 

OF THE DEGREE. 

If a degree is 69£ miles, the earth's diameter is 7960§£ 

360 
G9£ 

3240 
2160 
180 

22 ) 25020 ( 7960ff 
7 



175140 
154 

211 

198 

134 
132 



As it is not convenient for me to find the diameter of the 
earth by astronomy, I have to take it from books, and they 
vary ; but by the plan I have taken I make it 7906 English 
miles, which is 1 mile more than Hodson makes it ; so I call it 
right. Now, by Archimedes' proportion, as 7 to 22, it makes 
the diameter 7960^. 

I make the circumference, 25020 
Diameter, 7901 



25020 
225180 
175140 

197683020, square miles. 



THE MEASURE OF THE CIRCLE, 35 



Mr. Hodson makes his 

circumference, 24840 

7900 



22356000 

173880 



196236000 
Now, I subtract Mr. Hodson's from mine : — 

197683020 
196236000 



1447020 



In squaring the circle I find in all cases, after measuring the 
6 oblong squares, equal to 3 times the square of the radius, 
which is all that may be expected. The remainder is the 
biquadrate, which is termed segments, which in 12 inches 
diameter is just 4 inches, which added to the square inches, 
108, which is 3 times the square of the radius, completes the 
area, 112. It is my opinion that, as in a round table the 
biquadrate measures the circle, or the circular part, it is when 
squared what is not improperly called squaring the circle. 

The square root of 4 is 2 ; therefore the square of the circle 
12 inches in diameter and 38 in circumference is 2 inches, 
which added to 3 times the square of the radius, or 3 times 
the diameter, makes 38, circumference. But the square of the 
area is 10.583^5-- 

THE DISPUTE. 

My measure of the circle has been disputed, by saying my 
way would not be right in any number but 12 ; for by taking 
half the biquadrate the area would not be the same. This I 
agree to ; but I never said take half the biquadrate, but take 
the contents of the biquadrate, which in 12 diameter would be 
4, and add it to 3 times the square of the radius, and it would 
be the complete measure of the area, equal 112; and take the 



36 THE MEASURE OF THE CIRCLE. 

square root of the biquadrate, equal 2, and add it to 3 times 
the diameter, and it would complete the measure of the circle. 

I am aware that though the square root of 4 is 2, equal to 
half the contents, the square root of 16 would be but 4, equal 
to one fourth the contents. 

The straight line of a circle, or 6 sides of the hexagon, 
measures just 3 times the diameter, so that the circumference 
of 12 diameter would be 36 inches. But in curying that line 
to a circle, every sixth of an inch gains -^ on a straight line, 
and twice that on a circle, which is equal to $$, as proved by 
mathematics ; so that 6 inches on a straight line is equal to 6J 
on a circle ; but no instrument made with hands can demon- 
strate this measure. 

I find, by examining different works, that people try to find 
the area of the circle by trigonometry, which is very imperfect, 
for its result is square measure, the angle of which is in the 
proportion of 90 degrees, whereas the angle of a circle is in 
the proportion of 60 degrees. 

If 90 : tf : : 60, answer *jfo» Now, in a circle 12 inches in 
diameter, and whose radius is 6 inches, 3 times the square of 
the radius equals 108 ; divide 108 by 27, equal 4, which added 
to 108 equals 112, area. 

But by trigonometry, if 90 : tV : : 60, it gives ^ f° r angle ; 
and ^ of 12 is 6, which added to 108 equals 114, area; 
making the segment that you cut off from the oblong square 
near } of the square itself, whereas it should be but £ ; for 
there is not room in the circle to contain more. So you may 
see by the above that the circle gains T V, which in 12 diameter 
is just 2 inches ; for as one side of the hexagon is 6 inches on 
a straight line, 6 times 6 are 36, which divided by 18 equals 2 
inches, which I gain by curving the straight line to a circle. 
So 36 and 2 added equal 38, circumference. 

So the circle gains 2, and the square of 2 is 4, the square 
of the circle, independent of the dodecagon, or polygon of 12 
sides. And 4 is also the 27th part of 3 times the square of 
the radius, equal to 108 inches, as seen above. And 4 is equal 



THE MEASURE OF THE CIRCLE. 37 

to the square of the biquadrate, thus : 3 ) 108 ; 3 ) 36 ; 3 ) 12 (4. 
So if I add ^r to the straight lines of the rings, I make just 
the right area for them ; and, consequently, the measure of the 
circle must be right, because the area will just fill the circle. 

LEARNED MEN. 

I have been told by great men that the measure of the circle 
was found as near as it ever could be, and it was near enough 
for any thing ; but I find that, by the present measure in use, 
the convex surface of the earth Mis short of the true measure 
1.160107 geographical square miles. Dividing 1473336004 
by 127 produces for answer 1.160107. Circumference, 21600 ; 
multiplied by 6821 produces 147333600. So it seems to me 
that correct measure will do no harm, for it is a good principle 
to know and practise the truth. 

To prove that no measure is perfect without the perfect 
quadrature of the circle, I take the table of long measure, and 
prove by the great circle of the earth, which is so many miles, 
yards, feet, &c, that the carpenter's rule of 12 inches is near 
T V of an inch too short, and the yardstick near T 3 g- too short, and 
some square measure is still further astray. But some of our 
great men say it is near enough for any thing. Astronomers 
may pronounce it such, for if they are 10 or 20 millions astray, 
the vulgar know no better than to believe them ; but if a me- 
chanic made his joints lack as 7 to 22, we should have large rat - 
holes in our buildings, and some of our learned men might be 
eaten up. 

I received a letter from a professor of high standing, stating 
that it was discovered by mathematicians that if the diameter 
was 1, the circumference was 3.14159; so I work by that, and 
prove the difference, thus : — 

3.14159 
12 



37.69908 
4 



THE MEASURE OF THE CIRCLE. 



My circumference is 38 ; so I substract from 

3800000 
3769908 



.30092 too little. 

So you see it is T 3 <j and Tulroiixr too short in a circle 12 inches in 
diameter. To find what part it is too short in measure, divide 
the circumference by the difference : 30092 ) 3800000 ( 122956 ; 
divide the circumference by 122956: — 

122956)3800000(3 
368958 



11.132 



So you see it is very near ■£? to a yard short of measure. 

A circular staircase builder of New Haven — Mr. Bodsford 
— made a new circular stair rail, in the proportion of 1 to 
3.14159; and as I was passing, he asked me if I could tell 
him the length of that rail by figures. 1 said yes, and gave 
him the length. He said, " You are wrong ; it is only so 
long."* I said, "Then your rail is that much too short." 
"Yes," said he; "it is about that much too short." So, 
instead of the rail coming over the floor at the landing, it was 
li inches over the well hole. 

I have been told by mechanics that they had discovered that 
38 was the circumference of 12 diameter. 

SQUARING THE CIRCLE. 

To square the circle, in the first place adopt some certain 
diameter, say 12 inches. Now, I know that an angle of 90 
degrees is commonly called a right angle, but I think it might 
as well be called an erect angle, because it is erect from the 
horizon. It is a right angle for a square, but 45 is the right 
angle for a mitre, and any number of degrees would be the 
right angle for something. 



THE MEASURE OF THE CIRCLE. 39 

The reason I make this statement is, people tell me there is 
no right angle but that of 90 degrees, and that I must call it 
so, or otherwise they could not understand me. But I call 60 
degrees the right angle to measure a circle, and the only right 
one. Now, the groundwork to measure a circle is, 6 triangles 
equal to 18 perfect angles ; and there is no number but 60 that 
will make them perfect ; and if they are imperfect, my measure 
will be imperfect £ of £, equal to T V. 

In the first place, to the 6 sides add &, equal 2, in propor- 
tion of 90, linear measure, for the circumference. But ^V* 
although right in the linear, is wrong in the artificial. Now, 6 
inches being the radius, T \ is equal to 2, and the square of 2 
is 4, which is the multiplier, and 2, equal £ the angle, is the 
divisor. Now, there are 18 angles, of 60 degrees each; there- 
fore £ of T V equals 54, divided by 2 equals 27. Now add the 
4, which in the proportion of 90 degrees is yV> but in the pro- 
portion of 60 is sY, thus : — 

27 

1 



112, area. 
And the circumference is 38. 

To make it more plain, I go through another operation, and 
suppose 6 to be the diameter. Now, 18 angles, radius 3 inches ; 
take 6 and \ of £, equal to T V, add to 6, equal 19, circumfer- 
ence. And of 18 angles and £ of ^, equal 54, divide 54 
by 2, equals 27 ; add 1, equals 28 ; multiply by 1 gives 28, 
area, and 19 inches, circumference. 

I was told by an English gentleman, a lawyer, that there 
was no measure that was correct, for the want of a perfect 
quadrature of the circle ; and experience has taught me that 
his words were true. By examining the table of long measure 
I find that in proportion as the circle is measured wrong, every 



40 THE MEASURE OF THE CIRCLE. 

other measure must and will be wrong ; for the measure-of the 
circle is what all measure springs from ; it is the basis of all 
measures and weights. 

Now, for instance, I have taken Archimedes' measure, as 7 
to 22, which is the nearest to correct measure that I have ever 
found, and it is just J of an inch in 33 too short, almost T 3 ^ in 
a yard, and T V in a foot ; and every other 'measure will be astray 
in the same proportion. 

To find the area of a circle : On a subsequent consideration 
on finding the area of the circle by 3 times the square of the 
radius, I was thinking that sometimes the radius would be a 
hard number to square, as in a case where a surd number comes 
in ; and as it is considered impossible to perfect the square of a 
surd number, I will give another way of finding the area, thus : 

Multiply 3 times the radius, and the result will be the same 
as 3 times the square of the radius. Suppose 12 the diameter, 
6 the radius ; then 3 times 6 equal 18 ; 

18 



27 ) 108 ( 4 
108 



Now add the 4 to 108, which gives 112, area. 

To find the measure of a circle : 60 degrees is the only per- 
fect triangle, and 6 is the only perfect number ; so I take 6 of 
these triangles, and they form a hexagon, which is the basis of 
a circle, and there will be 6 sides and 6 radii. Then, to obtain 
the circumference, I square the radii by 6 ; 6 X 6 = 36, de- 
scending square. Then square the square : — 

36 
36 

216 
108 

1296 



THE MEASURE OF THE CIRCLE. 41 

The biquadrate is equal to 4k of -sir ; and the sides of the, 
hexagon being divided by 36, the two sides of the biquadrate 
will equal -£$ or T V of the side of the hexagon. So, as the 
biquadrate gains T V on one side of the hexagon, it will gain T V 
of the 6 sides ; therefore to 36 add T V, equal to 2, which gives 
38 for circumference. 

THE RATIO FOR A CIRCLE. 

In regard to finding a correct ratio, I was told by a professor 
in Connecticut that I never could do it, for I might figure half 
way round the earth, and it would never come right. I was 
told about the same by a professor in London. His reason was, 
because it was a surd number, which I allow has been called an 
impossibility. This gentleman in London, to convince me that 
it is impossible, takes his pencil, and divides 19 by 6, thus : — 
6 ) 19 ( 31666, and says it will always be 6, which proves it to 
be an impossibility, and throws away his pencil. I knew that 
this was one of the greatest difficulties — to find a correct ratio 
— and to do it I had to find a correct way to work out the surd 
number, which I have laid down in my work, as showing how 
I came by my ratio. 

As I have before said, the measure of the circle is the only 
perfect and independent measure, from which all other meas- 
ures spring, and without which no measure can be proved or 
worked right. We find in the table of long measure that 3 
barleycorns make an inch. This is imperfect, as the length of 
a barleycorn is not known. As 7 to 22 is the nearest correct 
ratio that has come to my knowledge, and this is | of an inch 
short of measure, the cubic inches in a yard would be 46656 ; 
but by the ratio 6 to 19 there would be 47724 ; so it lacks yfe 
in length, equal to 553^-$$; and as there are 3 sides to a cube, 
I must multiply by 3 : — 



42 



THE MEASURE OF THE CIRCLE. 



553^ 
3 

1060 78 

8, add for angle. 

1068, short of measure. 

46656 
1068 



47724.78, correct measure. 



So the present measure falls short as follows : Linear meas- 



ure, 



75f 



per cent., square measure, 1.52s- P er cent., cubic 
measure, 2.29 per cent. 

To find the circumference of a circle 12 inches in diameter : 
Take the hexagon, composed of 6 triangles, having 18 angles. 
Now, one angle on the hexagon is 6 inches ; so I take 2, equal 
to T V of the whole, and of that I make an oblong square, 6 
inches long by 2 inches wide, with a line through the centre. 




The two sides of the oblong square represent the two radii ; 
and if a square be placed on the top, and drawn one side, and 
then drawn into the centre, it will form a cone, as is seen in the 
figure ; and the lines gained on the top will strike the sides of 
the oblong just T V above the top of the cone, which shows that 
a line drawn across the top of them to the two sides of the 
cone would be just yV longer than xV of the hexagon, which 



THE MEASURE OF THE CIRCLE. 



43 



would be just the measure of that part of the circle ; so it 
proves that the circle gains T V of the .hexagon ; and if the hex- 
agon equal 3 times the diameter, or 36, add T V, equal 2, which 
gives 38, the circumference. 

If the radius is 6 inches, then the sides of the square, by 
moving 1 inch to the centre point, in 6 inches will move just 
£ of an inch to the centre line, equal £ in the whole width. 
And 6 is equal to 18 thirds ; now to 18 add 1, = 19 ; twice 
18 are 36, and T V, equal 2, added to 36, equals 38, the cir- 
cumference. 

To prove that the angle to measure a circle should be 60 
degrees, instead of 90 : I find, by examining books on this 
subject, that they make the area of a circle 12 inches in diam- 
eter 114 inches, allowing it to be 38 in circumference. 1 
should make it the same if I wrought it by 90 degrees, as 3 
times the square of the radius, 6x6 = 36, X 3 = 108 ; add 
6 to 108, =r 114. But I find the measure by the biquadrate, 
and that is but 2 inches ; the square of 2 is but 4 ; so 108 -f- 
4 — 112, instead of 114. So, if 90 give 6, 60 will give 4 ; 
add the 4 to 108, which gives 112. 

Proof. — Take an angle of the hexagon, one half of which 
is 3, and this, when divided, makes the length of leg only l£ ; 




so I make it all halves ; and as the radius is 6 inches, I mul- 
tiply by 12 halves : — 



44 THE MEASURE OF THE CIRCLE. 



2494 
400 


40000 
39786 


12 
12 

144 

9 Subtract 1 

135(11.6187 
1 

21 ) 035 
21 

226 ) 1400 
1356 




2094 
19 


0.0214 


i = 3, X 3 = 9. 


1.8846 
2.094 




3.9786 






2321 ) 4400 
2321 


60000 
5.8093 




23328 ) 203000 

185824 


0.1907, 1st leg. 
470, 2d leg. 
117, 3d leg. 

.2494 




232S67 ) 1807600 
1626369 



But as the outside of the oblong square and the polygon is 
taken 5 times, and should be taken but once, it is y^ too 
much, which must be subtracted. Then multiply the remain- 
der by 19, equal one half the perimeter, and it falls short of 4 
inches by t§wtt j but the polygon f long, &c, will make it 
good. So you see that 112 is the area of the perimeter; and 
if the area is right, when made into rings, as before shown, the 
measure of the circle cannot be wrong. 

I have measured the circle, and find it gains just the biquad- 
ate ; but there are few that can find the perfect roots ; if they 
could, they would have saved me much trouble, in respect 
to my problems ; for if they were masters of the square root 
they might soon see that they could not get 113 T V inches within 
a circle of 12 inches diameter and 37 T V perimeter, when I can 
get only 112 in one of 12 inches diameter and 38 perimeter. 
See the diagram of rings on a previous page. The average 



I 



THE MEASURE OF THE CIRCLE. 45 

measure of the rings is 3^ inches, and they are 36 in number. 
Multiply 36 by '3£ =112, area ; and if this measure is correct, 
it is surprising that the measure of the circle should be wrong. 
Let common sense dictate ; I ask no more. 

Again : the groundwork of a circle is the hexagon, with 6 
triangles and 18 equal sides. Now, the radius is -J of £, equal 
iV ; and as every thing in a circle has equal proportions, the 
curve must therefore be equal ; and as the circle is the most 
perfect thing in nature, every part must have a natural and 
proportional result. The hexagon gives | of i for radius ; 
consequently, the radius must give % of ^ for curve, in every 
part; and £ of J is T V of itself; so the circle gains T V on the 
radius, and the radius is T V of the 6 angles of the hexagon, 
mark as you like ; and it will be the same as £ X i = tV \ an( l 
6 X 3 = 18 + 1 = 19, perimeter. 

TAKING THE HYPOTENUSE. 

As the proportion of 7 to 22, the most correct that has come 
under my observation, falls £ of an inch short of measure, I 
will show the number of cubic inches in a yard by this meas- 
ure, and by the true ratio, as 6 to 19 : — 

36 



216 
108 

of 132=44 ) 1296 ( 29.5 Add this 29.5 to 1296 
88 29.5 



416 1325.5, the square. 



200 7953.0 

39765 



47718, from as 6 to 
19. 



46 THE MEASURE OF THE CIRCLE. 

36 



216 
108 



1296 

36 47718, from as 6 to 19. 

46656, from as 7 to 22. 

7776 



1062, difference. 



46656, from as 7 to 22. 

This is worked by taking the hypotenuse of a circle. Suppose 
it is 12 inches in diameter ; first I take the radius, 6, for hypot- 
enuse ; one side of a polygon of 12 sides equals 3, and £ of 3 
is 1J, for known leg ; and as it comes halves, I multiply by 2, 
to bring it all into halves, saying twice 6 is 12, and 3x3,= 
9, subtracted : — 

12 

12 

144 
9 

135, as in the work. 

The second operation is but J of an inch on the circle ; there- 
fore, multiply by 4 : — 

6 

4 

24 



576 
9 

567, as in the work is shown. 



THE MEASURE OF THE CIRCLE. 



47 



12 
12 



144 
9 



21)135(11.61895 
35 
21 



3 

9, subtracted. 

2)11.61895 



5.80945 



226)1400 
1356 



600.000, radius. 
580.945 



2321)4400 
2321 



23228)207900 

185824 



19055 These two added together 
1000 make just 6 inches, 

and the oblong 



18055, 1st. square T thr too 
much. 



3706, 2d. 



232369)22076 
20913 



217610 
18* 



2323785 ) 



652830 
1.740880 
2.17610 l T 7 <y increase of circle. 



3.999263 toward the biquadrate of 
a circle 12 inches in 
The biquadrate is 4.000000 diameter. 

3.989895 toward 4,0. 



.000737, the lack of 4 inches in the 
biquadrate. 



I was called on to go to No. 356 Broadway, New York, to 
see what was called the squaring of the circle by two gentle- 
men from Cuba — Don Lewis de la Torrey Sages and Senor 
Sedano. They cut from a piece of metal — zinc or brass — a 
circle about 4 inches in diameter. They then cut this into 26 
pieces, and placed them in the circle where the piece was taken 
from, which filled the circle to perfection. They then knocked 



48 THE MEASURE OF THE CIRCLE. 

the 26 pieces promiscuously on the table, and took them one at 
a time, and formed with the pieces a perfect square. 

This is what I thought to he impossible. I think it was a 
most unexpected feat, and worthy of note and observation. 

This I claim as one of the convincing proofs of my measure 
of the circle, so far as mechanical operation can go to demon- 
strate it. 

My biquadrate, which is what makes up or measures the 
circle, is by them demonstrated to prove my measure. They 
cut the circular part, which is my biquadrate, to a square, 
which is evidence to the eye that my method is correct. 

Any person can see that if the circular part of a circle 12 
inches in diameter is cut off, so as to leave a perfect square, the 
difference in the circle obtained from as 7 to 22 and that ob- 
tained from as 6 to 19 will be clearly perceptible in the circu- 
lar part ; because the difference in a circle 12 inches in diam- 
eter, from as 7 to 22 or as 6 to 19, is T 3 o of an inch, which I 
think is perceptible to any one who looks with an honest eye. 

Notwithstanding all this, I claim that no mechanical demon- 
stration can prove the perfect measure of the circle, because 
mathematics will discover that point which the most discerning 
eye cannot discover. The mathematical evidence which I have 
exhibited in my work far exceeds any mechanical operation. 

THE ANGLES, ETC. 

A circle is a plane figure, which has an equal extension in 
every direction from its centre to its circumference. 

The diameter of a circle is a straight line passing through 
its centre, and extending in length to the extreme limits of the 
circle. 

The radius of a circle is a straight line drawn from the cen- 
tre to the circumference. 

The radius of a sphere is a straight line drawn from the cen- 
tre to the surface. 



THE MEASURE OF THE 



49 



An angle is the opening between 
two lines which, meet each other at a 
point. 



One straight line is perpendicular 
to another when the angles on each 
side of the perpendicular are equal to 
each other. 




Angles made by lines meet- 
ing each other perpendicularly, 
or crossings each other perpen- 
dicularly, are called right angles. 



An acute angle is one that is 
smaller or sharper than a right 
angle. 



An obtuse angle is one 
that is larger or more 
open than a right angle. 



A triangle is a plain figure, enclosed by 
three straight lines of circumference. 

An equilateral triangle is one whose three 
sides are all of equal length. 




50 



THE MEASURE OF THE CIRCLE. 




An isosceles triangle is one which has two 
sides equal to eac^L other. 




A scalene triangle is one whose 
three sides are unequal in length. 



A right angled triangle has one right 
angle. 

The hypotenuse is the longest side 
of a right angled' triangle, or the side 
opposite to the right angle. 



An obtuse angled triangle 
has one obtuse angle. 



All other triangles have three acute angles, and are called 



acute angled. 



A rectangle is a figure having four 
sides and four right angles. 




A square is a rectangle whose four sides are 
equal to each other. 



THE MEASURE OF THE CIRCLE. 51 



A rhombus is a plane figure, hav- 
ing four equal sides and two obtuse 
and two acute angles. 




A parallelogram is a plain figure with four sides, having the 
opposite sides parallel to each other ; and as parallel lines 
every where preserve an equal distance between them, the 
opposite sides of parallelograms are equal. 

An arc of a circle is any part of its circumference. 

The chord of an arc is a straight line joining the two extrem- 
ities of the arc. 

The segment of a circle is a part of the area cut off by a 
chord, or the part inclosed by an arc and its chord. 

An axiom is a self-evident truth, or one so manifest that it 
cannot be made more clear by any demonstration. fc 

The sign -f-, when placed after a number, denotes that more 
is to be added in order to complete the result. 

This sign — , called minus, when placed after a number, de- 
notes that something is to be taken from it.' 

This sign z=z, when between two numbers, denotes equality. 

This sign x denotes multiplication. 

This sign -7-, placed between two numbers or quantities, 
denotes that the one is to be divided by the other. 

This sign ^/ denotes that the square root is to oe extracted. 

This sign ^/ 3 denotes that the cube is to be extracted. 

A polygon is the general name applied to plane figures with 
any number of sides. 

A hexagon is a figure with six sides, equal in all its parts, 
different from all other figures, and will measure a circle, with 
the number 6. 



5£ THE MEASURE OF THE CIRCLE. 



MECHANICAL POWERS. 

The mechanical powers are certain simple machines used for 
raising greater weights, or overcoming greater resistance, than 
the natural strength of man can perform without them. 

These simple machines are six in number, viz. : 1, the lever, 
2, the wheel, 3, the pulley, 4, the screw, 5, the wedge, 6, the 
inclined plane. 

Force is a power exerted on a body, to move it ; if it acts 
instantaneously it is called percussion ; if it impels constantly 
it is an accelerative force. 

Gravity is that force which causes a body to fall downward. 
It is called absolute gravity when in an empty space, and rela- 
tive gravity when immersed in fluid. 

Specific gravity is the proportion which the weight of one 
body bears to another. 

The centre of gravity is a certain point in a body, upon 
which, when suspended, it will rest in any position. 

The centre of motion is a fixed point round about which a 
body moves ; and the axis of motion is that fixed line about 
which it moves. 

Power and weight, when opposed to each other, signify one 
body that moves another ; and the body that moved the body 
which communicates the motion is the power, and that which 
receives the motion is the weight. 

Friction is the resistance which any machine suffers by the 
parts rubbing against each other. In the operation of machines, 
though all bodies are rough, in some degree, and all engines 
imperfect, yet it is necessary to consider all planes as perfectly 
even, all bodies perfectly smooth, and all bodies and machines 
to move without friction or resistance ; all which sounds very 
plausible. 



THE MEASURE OF THE CIRCLE. 53 



ON MECHANICAL POWERS. 

The whole principle of relative motion in mechanics depends 
upon this one single rule : That the whole force of a moving 
body is the result of its quantity of matter multiplied by the 
velocity of its motion. 

Thus, when the products arising from the multiplication of 
the particular quantities of matter in any two bodies by their 
respective velocities are equal, the entire forces are also equal. 
For example : suppose a body which weighs 40 pounds to move 
at the rate of 2 miles in a minute, and another body, that 
weighs 4 pounds, to move at the rate of £0 miles in a minute ; 
the entire force with which these two bodies will strike against 
each other would be equal powers to stop them ; for 40 multi- 
plied by 2 gives 80, the force of the first body, and 80 is also 
the product of 4 multiplied by 20, the force of the second 
body. Therefore, the heavier any body is, the greater is the 
body or power required either to move or stop it ; and the 
swifter it moves, the greater is the force. When two bodies 
suspended on any machine are put in motion, and the perpen 
dicular ascent of one body multiplied into its weight is equal 
to the perpendicular descent of the other body multiplied into 
its weight, those bodies, however so unequal in their weight, 
will balance one another in all situations ; for as the whole 
ascent of one is performed in the same time as the whole 
descent of the other, their respective velocities must be directly 
as the spaces through which they move; and the excess of 
weight in one body is compensated by the excess of velocity in 
the other. Upon this principle the power of any machine may 
be easily computed ; for it is only necessary to find how much 
swifter the power moves than the weight does, that is, how 
much faster in the same time, and just in that proportion so 
much power is gained by the engine. 

The Lever is a bar, either of iron or wood, one part of 
which is supported by a prop at its centre of motion ; and the 
5* 



54 THE MEASURE OF THE CIRCLE. 

velocity of every part or point in the lever is directly as its 
distance from the prop. There are four kinds of levers : 1st, 
the common lever, in which, the prop is placed between the 
weight and power, but much nearer the weight than the power ; 
2d, when the prop is at one end, the power at the other, and 
the weight between; 3d, when the prop is at one end, the 
weight at the other, and the power is applied between them;" 
the 4th differs from a lever of the first kind only in being bent. 
Levers of the first and second kind are often used in engines ; 
the third kind is seldom used, as no power can be gained by 
them. When the power is at the same distance from the prop 
that the weight is, and the power and weight are both alike, 
the machine will remain in equilibrio, and no power can be 
gained. This is the principle upon which the common balance 
is formed. 

ELECTRICITY. 

The earth, air, and terrestrial bodies are supposed to contain 
a certain quantity of an elastic substance or subtile fluid, called 
by philosophers the electric fluid ; and when any body possesses 
more or less of this fluid than what naturally belongs to it, 
several effects are visible in it, and the body is said to be elec- 
trified. This certain quantity of electric fluid found in all 
bodies could never be increased or diminished if all bodies ad- 
mitted the passage of this electric fluid through their pores, or 
along their surface ; but there are many bodies which will not 
suffer this fluid to pass through them, .while others freely per- 
mit it. Those bodies through which the electric fluid can pass 
are called conductors of electricity, of which the most perfect 
are all kinds of metals. Those bodies through which the elec- 
tric fluid cannot pass are called non-conductors of electricity ; 
of which the most perfect are, glass, resin, sealing wax, sul- 
phur, beeswax, and baked wood, among solids, and oils and air 
among fluids ; but all substances become conductors when they 
are very hot. Conducting substances are also called non-el ec- 



THE MEASURE OF THE CIRCLE. 55 

trie, and non-conducting substances are called electric. Into 
these two classes all bodies are divided by electricians. 

When any body has acquired an additional quantity of elec- 
tric matter, and is surrounded with non-conductors, or bodies 
through which the electric fluid cannot pass, it must remain 
overloaded ; and if it has lost a part of its natural share of 
electric matter it must remain exhausted, because the bodies 
w^hich surround it prevent any of the electric fluid from enter- 
ing into or coming out of it ; and the body is then said to be 
insulated. 

There are two principal theories of electricity, each of which 
has had. its advocates. The one is, the two distinct electric 
fluids, repulsive with respect to themselves, and attractive of 
one another, adopted by Mr. Du Fay, on discovering the two 
opposite species of electricity, and since remodelled by Mr. 
Symmes 

PNEUMATICS. 

Pneumatics is that part of natural philosophy which treats 
of weight, pressure, and electricity of the air, with the effects 
arising from them. 

The air is that thin, transparent fluid body which surrounds 
the whole earth to a considerable height, and which, together 
with the vapors and clouds that float on it, is called the atmos- 
phere. That the air is a fluid is evident, from the following 
properties which it possesses, in common with all other fluids, 
viz. : 1, it yields to the least force impressed on it; 8, its parts 
are moved among one another ; 3, it presses according to its 
perpendicular height ; 4, and its pressure is every where equal. 

The reason the earth turns upon its axis is, the weight of 
air, there being 15 pounds' weight to every square inch upon 
the earth's surface ; that is, every square inch of earth sustains 
15 pounds' weight of air. The moon has no atmosphere; if 
it had it would turn upon its axis, the same as our earth, by its 
atmosphere, or weight of air. The moon does not attract but 
repels the water, which causes the ebbing and flowing of tides. 



56 THE MEASURE OF THE CIRCLE. 

If the moon attracted, like the sun, the water would be quiet 
and still at the full moon, instead of ebbing and flowing. But 
the moon repels and the sun attracts, which causes the ebbing 
and flowing of the tides. 

f ' By what way is the light parted which scatters the east 
winds upon the earth % }' — Job 38 : 24. Answer : In the first 
place, the weight of air on every square inch is calculated to 
be near 15 pounds. Now, I leave the college calculations, and 
take my experience as a mechanic. Suppose you are working 
machinery ; by a good regulator you will find that your ma- 
chinery takes more measure of water after sunrise than before. 
The reason of which is, the weight of the element is less on 
the surface, and it being more dense to the west, in the dark, 
than to the east, in the light, it rushes towards the east at the 
rate of more than 1000 miles per hour, and carries the earth 
round with it, which is what makes the earth's diurnal motion, 
and scatters the winds upon the earth. I think the wind blows 
more from the west than any other direction ; but if the above 
was not the case, it would be like to blow from the east more 
than 1000 miles per hour. 

This question has never before been answered, I think. 
When it seems to us a calm, the wind blows more than 1000 
miles per hour. 

CHANGE OF THE AIR. 

The following statement shows that the rarefaction of the air 
at a distance from the earth's surface increases in a geometrical 
proportion, while its height from the earth increases in an arith- 
metical proportion : 7 miles above the surface, the air is 4 times 
lighter; 14 miles, it is 16 times lighter; 21 miles, 64 times 
lighter ; 28 miles, it is 250 times lighter ; 35 miles, it is 1024 
times lighter ; 42 miles, it is 4096 times lighter ; and so on, in 
this proportion. From this it may be proved that a cubic inch 
of such air as we breathe on the surface would be so much 
more rarefied at the height of 500 miles that it would fill a 
sphere equal in diameter to the orbit of Saturn. 



THE MEASURE OF THE CIRCLE. 57 

The weight or pressure of air is determined thus : fill with 
purified quicksilver a glass tube about three feet long, and open 
at one end, and putting your finger upon the open end, turn 
that end downward, and immerse it in a small vessel of quick- 
silver, without letting in any air ; then take away your finger, 
and the quicksilver will remain suspended in the tube about 
29£ inches above the surface of that in the vessel. Therefore 
the air's pressure on the surface of the earth is equal to the 
weight of 29£ inches' depth of quicksilver all over the earth's 
surface, at a mean rate. A square column of quicksilver 29£ 
inches high and 1 inch thick weighs just 15 pounds, which, 
therefore, is equal to the weight of air upon every square inch 
on the earth's surface ; and the weight upon a square foot, 144 
inches, amounts to 2160 pounds. According to this, a mid- 
dling-sized man, whose surface is generally about 14 square feet, 
sustains a pressure of air of 30,240 pounds, when the air is of 
mean gravity. 

As the earth's surface contains near 200,000,000 square 
miles, in round numbers, and every square mile 27,874,400 
square feet, there are 5,575,680,000,000,000 square feet on the 
earth's surface ; which multiplied by 2160 pounds, the weight 
on each square foot, gives 12,043,468,800,000,000,000 pounds 
for the pressure of the whole atmosphere. 

All common air is impregnated with what is called the vivi- 
fying spirit, which is essential to preserve life ; and in a gallon 
of air there is enough of it for one man during the space of 
one minute, but not much longer. This spirit is also in the air 
which is contained in water, as appears by fishes dying when 
they are excluded from fresh air, as in a pond that is frozen 
over. 

This spirit in air is lost by passing through the lungs of any 
animal, which is the reason why an animal dies so soon when 
deprived of fresh air. The little eggs of insects, also, when 
stopped up in glass, and excluded from the air, do not produce 
their young, though they be assisted by warmth. The seeds 
of plants, though mixed in good earth, will not grow if deprived 
of air. 



58 THE MEASURE OF THE CIRCLE. 

The vivifying qualities are also destroyed by the air passing 
through -fire, particularly charcoal fire, or the flames of sulphur. 
Air may also become vitiated by being closely confined in any 
place for a considerable time, or by being mixed with malignant 
vapors ; and lastly, by being corrupted by vitiating spirit, as 
in the holds of ships, or in oil cisterns and wine cellars which 
have been shut for some time, or in brewers' vaults. In any 
of them the air may be so much vitiated as to cause immediate 
death to any animal that enters. 

When the air has lost its vivifying spirit it is called damp, 
because it abounds with humid and moist vapors, and destroys 
life. This is known by those who work in mines. 

When a part of the vivifying spirit of the air in any coun- 
try begins to putrefy, the inhabitants of that country will be 
subject to an epidemical disease, which will rage till the putre- 
faction is over. And as the putrefying spirit occasions the 
disease, so if the diseased bodies contribute towards the putre- 
faction of the air, the disease will become pestilential and 
contagious. 

THEORY OF THE WINDS. 

Wind is the consequence of the rarefaction of air, and is no 
other than air put into motion by heat, or any other cause ; for 
when the air is rarefied by heat it will swell, and thereby affect 
the adjacent air; and the degrees of heat being different in 
different places, there will arise various winds. When the air 
is heated to any degree it will ascend upward, and the adjacent 
air will rush in to supply its place ; therefore there will be a 
stream or current of air from all the adjacent parts towards the 
place where the heat is. This appears evident from the motion 
with which the air rushes towards any place where there is a 
great fire, as in a glass house, or through the key hole of a 
door in a room where there is a fire. 

That wind called the trade wind, which blows constantly 
from the east to the west, about the equator, is a necessary con- 
sequence of this principle ; for when the sun shines perpen- 



THE MEASURE OF THE CIRCLE. 59 

dicularly upon any part of the globe, the air in that part will 
be heated, and, consequently, rarefied, and will therefore rise 
up. When the sun withdraws, the adjacent air rushes in to 
fill the place of the rarefied air, which will cause a stream or 
current of air from all parts towards that part which is most 
heated by the sun ; and the course of the sun being from east 
to west, with respect to the earth, the common course of the 
air which supplies the place of the rarefied air must be in the 
same direction, viz., from east to west ; but on the north side 
its course will be directed a little towards the north, and on the 
south side as much towards the south. 

This would be the general cause of the wind about the 
equator, if it were not affected by other causes which change 
its direction, as, 1st, by exhalations that arise out of the earth 
at different times and different places, occasioned by subterra- 
neous fires, volcanoes, &c. ; 2d, by a sudden inundation of rain, 
which causes an extraordinary rarefaction of the contiguous air ; 
3d, by high mountains, which change the course of the winds ; 
4th, by the declination of the sun towards the north or south, 
thereby causing a greater heat in the air on the same side of 
the equator. 

The following are the principal causes which create such a 
great variety and uncertainty of the winds in most countries 
distant from the equator : 1, the variations of the winds in 
different parts of Europe ; 2, the monsoons, which are found in 
the Indian seas ; 3, those winds which always blow from west 
to east on the western coasts of America, and on the east of 
Guiana, and the sea breeze which in hot countries blows from 
sea to land in the day time, and the land breezes which blow 
towards the sea in the night time, and all others. 

THE CAUSE OF THUNDER AND LIGHTNING. 

The effluvia and vapors arising from different bodies meet 
and unite together in the atmosphere, which is the common 
receptacle of all vaporous bodies, as the steam from most 
bodies, the smoke from bodies burned, and the effluvia emitted 



60 THE MEASURE OF THE CIRCLE. 

from sulphurous, nitrous, acid, and alkaline substances, and 
every volatile body, rises to a certain height in the atmosphere, 
according to its own specific gravity ; and when the effluvia 
which arise from an acid and alkaline body meet each other in 
the air, there will be a conflict between these two vapors, or 
what is commonly called fermentation, between them. If this 
fermentation be great, it will produce fire ; and if the effluvia 
be of a combustible nature the fire will run from one part of 
the air to another, following the inflammable matter. 

These things may be demonstrated by the following experi- 
ment: Mix some oil of cloves and Glauber's spirits of nitre 
together, which will immediately produce a sudden fermenta- 
tion, with a fine flame ; and if the ingredients be wet, there 
will be a sudden explosion. These are the effects of the union 
of an acid and alkaline fluid. 

From this experiment we may account for the effects of 
thunder and lightning, which are occasioned by the effluvia of 
sulphurous and nitrous bodies meeting each other in the air, 
where, assisted by the sun's heat, a fermentation, fire, and ex 
plosion ensue. When the inflammable matter is thin and light 
it will ascend to the upper parts of the atmosphere before the 
fermentation, fire, and explosion take place ; but when it is 
more dense it will hover round the surface of the earth, where, 
when an explosion takes place, the fire is visible, and often dan J 
gerous ; the explosion also has a violent force, and the heat, 
being great, will rarefy and drive away all adjacent air, kill 
men and cattle, split rocks and trees, &c. Lightning differs 
from all other fire ; it has been known to pass through leather, 
wood, cloth, and other substances, without heating them ; and 
at the same time melting iron, steel, silver, gold, and hard 
metals and bodies. It has melted or burned asunder a sword, 
without hurting the scabbard, and melted money in a man's 
pocket, without hurting him or his clothes. So fine are the 
particles of this fire that they pass through soft, loose bodies, 
without injuring them, and spend their force upon those more 
dense. Any steel instruments, as knives and forks, &c, which 



THE MEASURE OF THE CIRCLE. 61 

have been struck with lightning, have a strong magnetic virtue, 
which they retain for many years. The lightning has often 
turned the magnetic needle round, and made it point to the 
south pole, instead of to the north. 

The explosions which sometimes happen in mines, and are 
called fire damps, are of the same nature with lightning, and 
are occasioned by sulphurous and nitrous vapors rising from the 
mine, which, mixing with the air, take fire from the light used 
in the mine. This fire, when once kindled, continues to run 
from one part of the mine to another, wherever the combusti- 
ble happens to be ; and as the electricity of the air is increased 
by the heat, the air in the mine will expand, and, for want of 
room, will explode, with a degree of force equal to the violence 
of the fire, the quantity of effluvia, and density of the vapors. 
This is sometimes so strong as to blow up the mine ; at other 
times so weak that when it has taken fire it may be easily blown 
out. Air that will take fire from the flame of a candle may be 
produced thus : Having pumped the air out of the receiver of 
an air pump, let the air run into it through the flame of the 
oil of turpentine ; then move the cover of the receiver, and 
holding a candle to the air, it will take fire. 

"When combustible vapors are kindled in the bowels of the 
earth, where there is little or no vent, they produce earth- 
quakes, and as soon as they break forth into the open air, vio- 
lent storms, or hurricanes of wind. An artificial earthquake 
may be produced thus : Take 10 or 15 pounds of sulphur and 
an equal quantity of the filings of iron, and knead them with 
common water into the consistency of paste ; this being burned 
under ground will in 8 or 10 hours' time burst into flames, 
and cause the earth, to tremble around it to a great distance. 
It is owing to substances of this kind and nature that we have 
volcanoes. 

HYDROSTATICS. 

Hydrostatics treats of the equilibrium of fluids, or the gravi- 
tation of fluid bodies remaining at rest. When this equilib- 
6 



62 THE MEASURE OF THE CIRCLE. 

rium is removed, and the fluid body set in motion, the effects 
that it then produces are called hydraulics. 

The siphon is a bent tube. 

A valve is a kind of flap or cover fixed to a pipe, or to the 
aperture of any body, and which, by opening only one way, 
suffers water or any fluid body to pass, but not to return. 

A piston is a small cylinder fixed to the end of a rod, and 
fitted to the bore of a pipe, and frequently contains a valve. 

AXIOMS. 

All fluids except air are incompressible, or incapable of being 
compressed into less space or shape. 

In a vessel of water, or any other fluid body, the pressure 
of the upper parts on the lower is in proportion to the depth, 
and is the same at the same depth, whatever the diameter of 
the vessel may be. 

The pressure of a fluid upward is equal to the pressure 
downward at any given depth. 

The bottom and sides of a vessel are pressed by the fluid it 
contains in* proportion to its height, without any regard to the 
quantity. 

If fluids of different gravities be contained in the same ves- 
sel, the heaviest will be at the bottom, the lightest at the top, 
and the others farther distant from the top, in proportion to 
their respective gravities. 

The direction of the pressure of a fluid against the sides of 
the vessel that contains it is in lines perpendicular to the sides 
of such vessel. 

A body that is heavier than an equal quantity of fluid will 
sink in that fluid ; but if lighter it will swim at the top of the 
fluid ; and if it be of the same gravity it will neither sink nor 
swim, but will remain suspended in any part of the fluid. 

A solid immersed in a fluid is pressed on all sides by the 
fluid in proportion to the height of the fluid above the solid ; 
and bodies very deeply immersed in any fluid may be consid- 
ered as equally pressed on all sides. 



THE MEASURE OF THE CIRCLE. 63 

Every solid immersed in a fluid that is specifically lighter 
loses as much of its own weight as is equal to the weight of a 
quantity of that fluid of the same dimension with the solid. 

The fluid in which the solid is immersed acquires the weight 
the solid loses. 

As the principal fluid with which we have any concern in 
hydrostatics is water, it may be necessary to name a few of its 
distinguishing qualities or properties : — 

1. "Water is a transparent, colorless, scentless fluid, which, 
with a certain degree of cold, turns to ice. 

2. "Water is one of the constituent parts of all bodies, as has 
been proved by distillation; for the earth, bones, the driest 
wood, and stones pulverized constantly yield a certain quantity 
of water. 

3. Though fluidity is commonly regarded as an essential 
property of water, yet many philosophers, particularly Dr. 
Black, of Edinburgh, consider it as an adventitious circum- 
stance, and produced by a certain degree of heat ; and they 
assert its natural state to be crystalline, as when frozen. 

4. "Water is a more penetrating body than ah', though it be 
less transparent ; for it will pervade bodies that air will not, as 
is evident from its passing through the pores of a bladder. 

5. Some bodies are dissolved by water, as salts, while it con- 
glutinates others, as brick, stone, and bones. 

6. Water owes its fluidity to heat, and it contains no small 
quantity of air ; and the sediment found in all water which has 
not been distilled always contains a quantity of earth, from 
which last element it is supposed that plants derive all their 
nourishment. 



64 



THE MEASURE OF THE CIRCLE. 



A TABLE OF SPECIFIC GRAVITY. 



Platina, . . 
Fine gold, 
Standard gold, 
Lead, . . . 
Fine silver, . 
Standard silver, 
Copper, . . 
Copper halfpence, 
Gun metal, . 
Fine brass, . 
Cast brass, . 
Steel, . . . 
Iron, . . . 
Pewter, . . 
Cast iron, 
Tin, . . . 
Lapis calaminaris, 
Loadstone, . 
Mean specific gravi 
the whole earth, 
Crude antimony, 
Diamond, 
Granite, . 
"White lead, . 
Island crystal, 
Marble, . . 
Pebble stone, 
Kock crystal, 
Pearl, . . . 
Green glass, . 
Flint, . . . 
Common stone, 



ty of 



1. — Solids. 

23000 Crystal, 2210 

19640 Clay, 2160 

18888 Oyster shells, .... 2092 

11325 Onyx stone, .... 2092 

11091 Brick, 2000 

10535 Common earth, . . . 1984 

9000 Nitre, 1900 

8915 Vitriol, 1880 

8784 Alabaster, . • . . . 1874 

8350 Horn, 1840 

8000 Ivory, 1825 

7850 Sulphur, 1810 

7645 Chalk, 1793 

7471 Solid gunpowder, . . 1745 

7425 Alum, 1714 

7320 Dry bone, 1600 

2000 Sand, 1520 

5000 Lignumvitae, .... 1327 

Coal, 1250 

4500 Jet, 1238 

4000 Ebony, 1177 

3517 Pitch, 1150 

3530 Rosin, 1100 

3160- Mahogany, .... 1063 

2720 Amber, 1040 

2705 Brazilwood, . . . .1031 

2700 Boxwood, 1030 

2650 Beeswax, 955 

2630 Butter, 940 

2600 Oak, 925 

2570 Gunpowder, shaken, . 922 

2500 Logwood, 913 



THE MEASURE OF THE CIRCLE. 



65 



TABLE OF SPECIFIC GRAVITY, CONTINUED. 

Ice, 908 Eir, 

Ash, 800 Sassafras wood, . 

Maple, 755 Cork, .... 

Beech, 700 New fallen snow, 

Elm, 600 



550 

482 

240 

86 



Quicksilver, . 
Oil of vitriol, 
Oil of tartar, 
Honey, . . 
Spirits of nitre, 
Aqua fortis, . 
Treacle, . . 
Aqua regia, . 
Human blood, 
Urine, . . . 
Cow's milk, . 
Sea water, 



2. — Fluids. 

13600 Ale, x . . 1028 

1700 Vinegar, 1026 

1550 Tar, 1015 

1450 Common water, . . . 1000 

1315 Distilled water, ... 993 

1300 Red wine, 990 

1290 Proof spirits, .... 931 

1234 Olive oil, 913 

1054 Pure spirits of wine, . 866 

1032 Oil of turpentine, . . 800 

1031 Ether, 726 

1030 Common air, . 1.232, or 1& 



As these numbers are the weight of a cubic foot, or 1728 
cubic inches, of each of the foregoing bodies, in avoirdupois 
ounces, the quantity in any other weight, or the weight of any 
other quantity, may be found by proportion. For example : 
required the contents of an irregular block of common stone, 
weighing 100 pounds, or 1792 ounces. Here, as 2500, the 
ounces in a cubic foot of common stone, is to 1792, so is 1728, 
the inches in a cubic foot, to 1238| cubic inches, the contents. 



THE DIVISIONS AND SOUNDS OF LETTERS. 

In the early ages of antiquity, before alphabets were invent- 
ed, mankind, sensible of the want of some means of recording 
events, and historical and scientific discoveries, had recourse to 
various arts for these purposes, the first of which was painting 
6* 



66 THE MEASURE OF THE CIRCLE. 

That partiality for pictures, so evident in all ages and countries, 
afforded the ancients a method of perpetuating their actions. 

To commemorate that one man had killed another, they 
painted the figure of a dead man, with another man standing 
over him, with a hostile weapon in his hand. 

On the first discovery of America, this was the only kind of 
writing used by the Mexicans. 

The first improvement by our ancestors in the art of writing 
was by the introduction of hieroglyphical characters. These 
consisted of certain symbols, which were made to represent cer- 
tain invisible objects. An eye was the symbol of knowledge ; 
a circle was eternity, as having neither beginning nor end. The 
figures of animals were much employed in this kind of writing, 
on account of some quality of which they were supposed to be 
endowed, and in which they resembled the object signified. 
Thus imprudence was represented by a fly, wisdom by an ant, 
victory by a hawk. 

These hieroglyphics flourished most in ancient Egypt, as did 
all other learning at that time, when the knowledge of these 
characters was reduced into a regular art ; and they still exist 
in Egypt, in some degree. 

The Chinese still use characters of this nature. They have 
no alphabet of letters, but every single mark or character signi- 
fies one perfect idea or object. The number of these characters 
is seventy thousand ; and to make acquaintance with them would 
constitute a labor for the whole life of one man. 

The next improvement in the art of writing was by the inven- 
tion of signs or marks which stood not directly for the objects 
themselves, but for the words or names whereby they were dis- 
tinguished. This was an alphabet of syllables. An alphabet 
of this kind is still in use in Ethiopia and some countries in 
India. 

It has been considered that the discovery of the earth's trav- 
erse round the sun was new ; but I find by Hodson that these 
systems have now given place to that called the Copernican 
system, which, undoubtedly, is the most ancient in the world 



THE MEASURE OF THE CIRCLE. 67 

It was first introduced into Greece and Italy by Pythagoras, 
and from him called the Pythagorean system. It was adopted 
by Philolaus, Plato, Archimedes, and all the most ancient phi- 
losophers, but was at length lost under the Peripatetic philoso- 
phy, and restored again about the year 1500 by Nic. Coper- 
nicus. 

This system has been proved, by the most evident demonstra- 
tions, to be the only true one. First is a representation of this 
system, where the seven concentric circles marked Mercury, 
Venus, the Earth, Mars, Jupiter, Saturn, and Georgium Sidus, 
represent the orbits of these seven primary planets, each per- 
forming its annual rotation round the sun, which is placed in 
the centre. 

The next two circles represent the twelve signs of the zodiac, 
with all its divisions into thirty degrees in each sign; and, 
lastly, the next outer circles show the twelve calendar months 
of the year, with their divisions into days, each in its order. 

TO CORRECT MEASURE. 

As the foot is divided into 12 inches, so each inch is divided 
into 12 parts, called seconds, and each second is again divided 
into 12 thirds, and each third into 12 fourths, &c. 

Now, according to correct measure, derived from a perfect 
quadrature of the circle, being in proportion of 6 to 19, instead 
of 7 to 22, which makes a difference of as r^ is to y^, the 
foot rule, which is called 12 inches, is near T V of an inch too 
short ; so it will have to be made near T V of an inch longer, 
and the T V or near ^ will be divided equally to each of the 12 
inches. 

By reason of the imperfection in the measure of the circle, 
every other measure, as well as all weights, are wrong in the 
same proportion. To rectify them, I take a stick or rod 6 feet 
long, and say, if 7 diameter gives 22 circumference, as 6 to 
19 gives 22£, so 3 times 22 are 5 feet 6 inches ; add 3 times 
£, equal to 5 feet 6J. The present measure, at 7 to 22, makes 
it on the rod 5 feet 6 inches ; and divide 5 feet 6£ inches into 



68 THE MEASURE OF THE CIRCLE. 

11 equal parts, equal to 5£ feet, which will be the correct 
measure in proportion as 6 to 19, from which you may take 
feet, inches, yards, &c, &c. And if the present measuring 
instruments are right, in proportion as 7 to 22, all the former 
tables will be right. 

With respect to weights, the gravity of a cubic foot of water 
is 62£ pounds, in proportion of as 7 to 22 ; but in propor- 
tion of as 6 to 19 it is 2.29 per cent, short; so the weights 
would require to be altered in the same proportion as the 
measures, and all would be right. The weights are full 36 
hundredths of an ounce in a pound short weight, by the pro- 
portion of as 6 to 19. 

MEASURES, ETC. 

Mensuration in general is the art of measuring and estimat- 
ing the magnitudes and dimensions of bodies or figures, and is 
divided principally into three parts, called linear measure, 
superficial measure, and solid measure. 

1. Linear measure is measuring length, without breadth or 
depth. 

2. Superficial measure consists of length and breadth taken 
together. 

3. Solid measure consists of length, breadth, and depth. 

4. A point has no parts nor diameter, neither length nor 
breadth. 

5. A line has only length, without any other dimension. 

6. A right line lies all in the same direction, and is the 
shortest way between its two extremities. 

7. A curve line continually changes its direction. 

8. Parallel lines are always at the same distance, and meet 
in no angle. 

9. Oblique right lines change their distance, and meet in an 
angle. 



THE MEASURE OF THE CIRCLE; by 

10. An angle is the meeting of two lines. 

11. If the two lines which form an angle be perpendicular 
to each other, they form a right angle. 

12. But if two lines be not perpendicular to each other, they 
form what is called an oblique angle, which is either greater or 
less than a right angle. 

13. An angle that is smaller than a right angle is called an 
acute angle. 

14. An angle that is greater than a right angle is called an 
obtuse angle. 

15. A triangle is a figure of three lines, and has various 
names, according to its angle. 

16. An equilateral triangle has its three sides, and conse- 
quently its three angles, equal to each other. 

17. An isosceles triangle has only two sides equal. 

18. A scalene triangle has its three sides and angles unequal 
to each other. 

19. A right angled triangle has one right angle. 

20. An obtuse angled triangle has one obtuse angle. 

21. An acute angled triangle has all its angles acute. 

22. A figure of four sides is called a quadrangle, or a quad- 
rilateral figure, and is either a parallelogram, a square, a rhom- 
boid, a rhombus, a trapezium, or a trapezoid. 

23. A square is an equilateral rectangle, having all its sides 
equal, and all its angles right angles. 

24. A rhomboid is an oblique angled parallelogram. 

25. A rhombus is an equilateral figure having all its sides 
equal, but its angles oblique. 

26. A trapezium is a quadrilateral figure, but its opposite 
sides are not parallel. 

27. A trapezoid has only two opposite sides parallel. 



$ 



70 THE MEASURE OF THE CIRCLE. 

28. Plane figures having more than four sides are generally 
called polygons ; but they receive particular names, according 
to their number of sides. Thus a polygon of five sides is 
called a pentagon ; one of six sides a hexagon ; one of nine 
sides a nonagon; one of ten sides a decagon; one of eleven 
sides an endecagon ; one of twelve sides a dodecagon. 

29. A circle is a plane figure, bounded by one circular line 
called the circumference, which is every where equally distant 
from the centre. 

30. The radius of a circle is a right line drawn from the 
centre to the circumference. 

31. The diameter of a circle is a right line drawn through 
the centre, and bounded at each end by the circumference. 

32. An arc of a circle is any part of the circumference. 

33. A chord is a right line joining the two extremities of an 
arc. 

34. The segment of a circle is any part of it. 

35. A semicircle is half a circle. 

36. A sector is a part of a circle contained under part of the 
arc and two radii drawn to the centre. 

37. A quadrant is a sector of a circle having one quarter of 
the circumference for its arc, and its two radii perpendicular to 
each other. ' 

38. The circumference of every circle, in geometry, is sup- 
posed to be divided into, 360 equal parts, called degrees, and 
each degree subdivided into 60 minutes, and each minute into 
60 seconds, and so on. Hence a semicircle contains 180 de- 
grees, and a quadrant 90 degrees, which form a right angle ; 
and half a quadrant, called an octant, contains 45 degrees ; for 
the measure of every right line angle is an arc of a circle con- 
tained between the two lines which form the angle, the point 
of the angle being in the centre of the circle ; and the number 



THE MEASURE OP THE CIRCLE. 71 

of degrees contained in the arc of the circle gives the measure 
of the angle. 

39. In every right angled triangle the side opposite the right 
angle is called the hypotenuse, and the other two sides the legs 
of the triangle. 

40. The height or altitude of a figure is a line drawn from 
the uppermost side or angle, perpendicular to the base. 

41. An angle is generally described by three letters, the 
name of a polygon. Thus, one of three sides is a trigon ; one 
of four sides a tetragon ; one of five sides a pentagon ; one of 
six sides a hexagon ; one of seven sides a heptagon ; one of 
eight sides an octagon; one of nine sides a nonagon; one of 
ten sides a decagon; one of eleven sides an endecagon; one 
of twelves sides a dodecagon. 

TO MEASURE A SPHERE OR GLOBE. 

Having given the measure of a circle, I will give a simple 
rule to measure a globe or sphere, having seen many errors in 
this measure, as no man could measure a globe without having 
the perfect measure of the circle. 

To find the true measure of the surface of a globe, in feet, 
inches, &c. : Suppose 12 is the diameter ; then the circumfer- 
ence will be 38 inches ; so multiply half the circumference by 
itself, and the product will be the true measure of its surface, 

thus : — 

19 
19 

171 
19 

361 

This is the true measure of the surface of a globe 1£ inches in 
diameter. 



72 THE MEASURE OF THE CIRCLE. 



THE PYRAMID. 

A pyramid is a figure that has a right lined figure for its 
base, and each of its sides is a triangle, whose vertices meet at 
a point at the top, which is called the vertex of the pyramid. 

The pyramid takes its name from the figure of its base, like 
the prism. So a pyramid is a square stick of timber, at a given 
square at the base, and running to a point at the top. 

To measure a pyramid : Square the diameter at the base ; 
take half that product; multiply that by the height, which 
gives the square feet. 

Suppose a pyramid be 100 feet high, and 4 feet square at the 
base, and run to a point ; how many square feet are there in it ? 

Square the base, 4 X 4= 164-8. 
4 

2)16 

8 
Multiply by the height, 100 

800 feet in the pyramid. 

It has been the custom to take the mean diameter of the 
base, which, if the base be 4, will be 2 ; and 2x2 = 4; and 
as it is 100 feet high, 100 X 4 = 400 feet, solid contents. But 
my way is thus : 4 feet square, base ; 4 X 4 = 16 ; then mul- 
tiply by the height, 100, = 1600 ; divide the 1600 by 3, which 
gives 533£ cubic feet. 

THE PRISM. 

A prism is a solid body, wood, stone, &c, as a stick of tim- 
ber, square at each end, and its sides alike. 

To find the measure of a prism : Multiply or square one 
end, and then multiply that product by the length, which will 
give the measure in cubic or solid feet. 

Suppose a square stick of timber to be 100 feet long, and 4 
feet square ; how many cubic feet will it contain ? 



THE MEASURE OF THE CIRCLE. 73 



16 

100 

1600 cubic feet in the above stick. 



THE CYLINDER. 

The cylinder is a round prism, being all its length circular. 

To measure a cylinder : Suppose a cylinder be 100 feet long 
and 1 foot in diameter ; how many square feet will it contain ? 
Find the area of one end, and multiply that by the length, 
which will give the answer. 



112 
12 



1728 ^ 134400 ( 77 feet, 9 inches, and yflfo. 
' 12096 



13440 
12096 



13440 
1296 

48, remainder. 

Or you may measure a cylinder in this way : Suppose your cyl- 
inder to be square, and as large as the diameter ; measure it the 
same as a square, and take $ of the product, which will give 
the circular measure. Thus, if your cylinder be 100 feet long 
and 1 foot in diameter : — 
7 



74 THE MEASURE OF THE CIRCLE. 



100 

7 



9) 700 (77 feet, 9 inches. 
63 

70 
63 



To measure a cone : Find the square of the base ; then mul- 
tiply by the height, and divide that product by 3, which gives 
the square contents ; then take J of the product, which gives 
the circular measure, thus : — 

16 

100 

3 ) 1600 

533J 

7 

9 ) 3733i 
414| *V. 

Suppose I had a stick of round timber to sell by the foot; 
the stick is 18 inches at butt, and 6 inches at the crop. I 
charge you one shilling a foot ; you buy half the length to-day, 
and the other half to-morrow; how much did you lose by 
buying it at two purchases ? 

Now, according to the adopted measure of timber of this 
description now in use, you lost ninepence. The operation of 
this sum, mathematically, will demonstrate the measure of the 
circle, and show its use. 

Suppose a round stick or block of timber 1 foot long and 17 



THE MEASURE OF THE CIRCLE. 75 

inches in circumference had to be squared ; how many inches 
would you lose in squaring ? Answer, 80£ inches. 

Multiply 2 and 6 pence by 2 and 6 pence. 

Answer, 6 and 3 pence. 
I have two half crowns, each equal to 5 shillings ; I want 
them multiplied so as to make 6 and 3 pence. 

"What does £1 19 s. lid. 3 qr. multiplied into itself amount 
to ? Answer, £ 3 19 s. 11 d. 3 qr. 

Subtract 19 rods, 5 yards, 1 foot, 5 inches from 20 rods. 

Answer, 1 inch. 

Suppose a ball be shot from a gun, on a level, 3950 miles, 
and then fall to the ground, perpendicularly; what distance 
must I travel to find the ball, and how far would it have to fall 
from where it lost its force ? 

Answer: I should have to travel 3126f miles, and the ball 
would fall 1636 T V miles, and a little over. 

Suppose I start from a centre, and travel on a radius 40 rods ; 
what length of chain, each link to be 1 inch long, or how many 
links would it require for the circumference ? 

Answer : It would require 5016 links. 

Suppose the reckoning of a party in a public house amounts 
to 6 shillings and 1 farthing ; what number of persons must 
there be, to pay an equal share ? 

Answer : IT persons, and each to pay 4£ pence. 

If i of 6 be 3, what will £ of 20 be ? * Answer : % 

What rule is that in multiplication which does not increase 
by multiplying ? Answer : Cross multiplication. 

Suppose a measure to be made in circular form, and to con- 
tain one bushel, allowing 2256 inches to the bushel ; what is 
the diameter and depth ? 

Suppose a circle to be 17 inches in diameter; what is the 
circumference and area ? 

Answer : The circumference is 53 inches, and the area 
224.77^ inches. 



76 THE MEASURE OF THE CIRCLE. 

What is the number of square feet or inches in a cylinder 22 
feet 6 inches long, and 4 feet 3 inches in diameter ? 

According to the measure now in use, from as 7 to 22, it 
would bring the world to an end in 66 years, which would be 
132 revolutions, thus : — 

22 
6 

2 ) 132 revolutions. 

66 years. 

As short measure forms an internal scroll, and over measure 
forms an external scroll. 

I have 36 straight strips, each 19 inches long and | of an 
inch wide. I wish to know how many square inches there are 
in each strip, and how many square inches there are in the 36 
strips, in proportion of square measure, and also in proportion 
of circular measure. When you work this, and understand it, 
you will see the difference between square measure and circular 
measure. 

Suppose a box be 2 feet long ; how wide and how high 
must it be to contain half a yard square ? 



Find the 


square 


of the 


length. 18 
18 

144 
18 

3)324 

108 

V 216 ( 14f£ 
1 

24)016 
96 



THE MEASURE OF THE CIRCLE. 77 

The box is 14} J by 24 long, 14f } wide, and 14f£ high. 
The above is not perfect. I work it thus : — 

18 

144 

18 

324 

18, to cube. 

2592 
324 



24 inches long. 24 ) 5832 ( 243 square inches. 
48 



103 
96 



72 
72 



*J ^43 ( 15.583, height and depth. 
1 

25 ) 143 
125 



305 ) 1800 
1525 



3108 ) 27500 
24864 



2636 



Suppose I have a round stick, 50 feet iong and 18 inches at 
the base, and running to a point. I wish to know the length 
of a string that is wound 3 times round each and every foot, 

7* 



78 THE MEASURE OF THE CIRCLE. 

commencing at the base and running to the extreme of the 12 
inches, the same as a barber's pole is painted. 

I find the circle, of the mean circumference, which is 9 
inches : — 

9 

9.5 



81 
4£ 



3 ) 85.5 ( 28.5, the circle. 
Now I square the ci*d.e. 28.5 

28.5 



1425 
2280 
570 



812.25 

This is what the string rises, 4 X 4 = 16 ; 

812.25 
16 



s/ 828.25 ( 28.77933 
4 



48)428 

384 



567 ) 4425 



5747 ) 45600 
40229 



57549 ) 537100 
517941 



575583 ) 1915900 
1726749 



5755863) 18915100 
17267589 



1647511 



THE MEASURE OF THE CIRCLE. 79 

28.77933 
150 



143896650 
2877933 



12)431689950(359 7.4162 
36 

"n 

60 



116 

108 

88 
84 



49 

48 

~19 
12 

~75 
72 

"To 

24 

~6 

Answer : The string will be 359 feet, 7.4162 inches. 

A and B bought 300 acres of land for 600 dollars ; each paid 
300 dollars ; A says to B, " Let me have my choice in the land, 
and my land shall cost me 75 cents an acre more than yours." 
The sum is 65 to 95. 



80 : 150 : : 
80 


90 
[ 126, answer 

remainder. 


80 
65) 

126, ans. 
184, ans. 

"iio 


: 150 : : 65 
80 


95 ) 12000 | 
95 

250 
190 


12000 ( 184, answer. 
65 

550 
520 


600 
570 

"30, 


300 
260 

40, remainder. 



80 THE MEASURE OF THE CIRCLE. 



310:126:: 
300 


300 
(121, 


ans. 


310 : 184 : : 300 
300 


310)37800 
310 


310) 55200 (178, ans. 
310 


680 
620 






2420 
2170 


600 
310 


2500 

2480 


290, 


remainder. 


20, remainder 



Answer: A has 121ff§ ; B has 178 B 2 T V 

To extract the circle or globe, after the manner of the square 
or cube root: Take any number of figures, say 448; divide 
the given number by 7, and add twice the quotient to the given 
number ; extract the square root of the same, and it will be the 
diameter. 



7 ) 448 ( 
42 


64 


64 

2 




448 • 
128 


28 
28 




128 




576 ( 24, diameter 
4 









44 


)176 
176 



To show the area of a circle, and its square : Cut a piece of 
wire 38 inches long, form a circle, and the diameter will be 12 
inches, and its area will be 112. Cut another piece 38 inches 
long, form a perfect square, whose sides will be 9j inches each, 
and whose area will be 90£ inches. You will see by this the 
difference of square and circular measure. The same length 
of wire formed into a circle measures in area 112 inches, and 
formed into a square, 90£ ; making a difference of £1} inches 
in the shape of the circle and square. 



THE MEASURE OF THE CIRCLE. 81 

There is a fish whose tail weighs 9 pounds ; his head weighs 
as much as his tail and half hrs body; his body weighs as 
much as his head and tail ; what is the weight of the fish ? 

Answer : His head weighs 27 pounds, his body 36 pounds ; 
the whole fish, 72 pounds. 

To measure a coal pit of wood : Suppose your pit is 60 feet 
in circumference ; first find the diameter ; and to find it divide 
the circumference by 19, and multiply the product by 6 ,* this 
gives the diameter. Then find the area, by taking J of the 
diameter ; that is the square of the diameter. Then multiply 
by the mean height, which will be 6 feet, as the pit is 12 feet 
high ; then you have the cubic feet. Then divide by the num- 
ber of feet in a cord, which is 128 feet, which gives the cords, 
feet, and inches. 

19 ) 60 feet, circumference. 



3.157 
6 



18.942 

So call the diameter 19 inches. 

19 
19 



171 
19 

361 This is the square. 

7 

I take J of the square. 9 ) 2527 

280 

This is J. I multiply the 280 by 6, half the' height. 

6 

1680 
I divide this by 128, the feet in a cord. 



82 THE MEASURE OP THE CIRCLE. 

128 ) 1680 ( 13 cords, 16 feet. 

128 



400 

384 



16 

Suppose your pit is 60 feet in diameter. 

60 
60 

3600 

7 



9 ) 25200 



2800 
6 



128 ) 16800 ( 131 cords, and 32 feet. 

THE SQUARE OF THE CIRCLE. 

The square of a circle 12 inches in diameter is 9 J inches. 
This is the square of the circle, but is not the square of the 
area of the same circle. 

The square of the area of a circle 12 inches in diameter is 
10 inches and a half, and T 8 jy, and a little over, thus : — 

112(10.583 

1 



205 ) 01200 
1025 



2108 ) 17500 
16864 



21163)63600 
63489 



111, remainder. 



THE MEASURE OF THE CIRCLE. 83 

Thus you see it is 10 inches, £, T 8 <j, and T -^. 
The fourth part of a circle is the square of the circle ; that 
is, the square of the circle itself, but not the area. Thus : — 

4 ) 38 ( 9.5 
36 

20 
20 



The circle gains over all other measure. Take a circle 38 
inches round, which is 12 diameter, and the area of this circle 
is 112 inches. Now take a square 38 inches round, that is, 
whose four sides measure 38 inches, and the area will be 90£ 
inches. 12 diameter is 112 area, 6 diameter is 28 area. It 
varies the same as the square ,• so that £ the diameter is £ in 
area. 

Suppose a circle 12 inches in diameter to be filled with rings 
I of an inch wide. Now, the length of the average ring is 19 
inches. Take 36 rings, 19 inches long and £ of an inch wide, 
and measure them by square measure, and you will make them 
measure 114 inches; but by circular measure you find they 
measure 112 inches. Now take 36 rings, 19 inches long and I 
of an inch wide, and by square measure (that is, 90 degrees) 
each ring will measure 3 inches and £, which will make the 36 
rings measure 114 inches ; but by circular measure, (that is, in 
proportion of 60 degrees,) each ring will measure 3 inches and 
£, and the 36 rings will measure 112 inches. 

This shows the difference between circular measure and 
square measure. It is in one riag T \ of an inch, which is cir- 
cular measure, in proportion of 90 degrees, which is to the 
square as 56 to 57; for twice 57 are 114, and twice 56 are 
112. The present measure is as 132, and correct measure is 
as 133. As 7 to 22 would be 132, and as 6 to 19 would be 
133. The difference between £ and £ is £. 

In regard to the taking of levels, you will find that great 



84 THE MEASURE OF THE CIRCLE. 

errors are made, for the want of a perfect quadrature of the 
circle. Thus I have frequently heard it said, if you make a 
platform perfectly straight and level, and place a perfectly round 
ball on it, it would make a perpetual motion. But they do not 
consider, I think, that the creation is on a perfect circle, there- 
fore a perfect level and a perfect straight cannot exist together 
in any one thing. In 16 miles it will get 56 yards out of 
level ; in 8 miles, 14 yards ; in 4 miles, 3} yards ; in 2 miles, 
2 feet 7 inches ; in one mile, 7§ inches ; in half a mile, 1£ 
inches ; in £ of a mile, f of an inclj. 

On a survey of the Isthmus of Darien, to consider the prac- 
ticability of cutting a passage through from ocean to ocean, to 
facilitate the navigation of the world, the return was, that one 
ocean was higher than the other, and therefore it would be 
impossible to cut such a passage. I think this return was 
wrong, for the want of the perfect quadrature of the circle ; 
for reason must dictate that these great bodies of water would 
naturally find their level, except agitated by winds, or some 
natural causes. 

In taking a level for a railroad, they may take a spirit level, 
and take a sight at a great .distance, which will be imperfect, if 
they do not calculate for the curve of the circle. 

I was asked by a man to give a plan to find a proportion for 
stove pipes, and other things of the kind, such as cabooses, &c. ; 
for which I take the rule of three, or proportion, thus : Suppose 
pipe to be square, say 6 inches ; then 6 X 6 = 36. Now, the 
circle is to the square as 7 to 9 ; so, by the rule of three, — 



If 9 


:36: 

7 

i 


:7 


9) 


252 






28, 


answer. 



Any other measure that may be required in your business 
will work in the same way. 12 X 12 = 144 ; so, — 



THE MEASURE OF THE CIRCLE. 85 



If 9 


:144: 

7 


:7 


9) 


1008 






112, 


answer. 



Thus you see the proportion may be got in a circle as per- 
fectly as in a square. Or suppose it is an oval ; then find the 
measure of the circle, as above, say 6 inches diameter : — 

If 9 : 36 : : 7 

7 

9)252 
28 

But say the oval is £ longer ; so 6 diameter, 7^ long ; the area 
of the circle is 28 ; so 4 times 7 are 28, which added to 7, = 
i, gives 35 inches for the area of the oval. 

The question is often asked, why a polygon of four sides 
will not measure a circle. There is no figure, of any number 
of sides, that will measure the circle, but the hexagon, or figure 
of six sides. This is the only figure that is equal in all its 
parts. The hexagon is from an angle of 60 degrees ; square 
measure is from an angle of 90 degrees ; these angles differ 30 
degrees. 

With the same propriety it might be asked, why will not an 
angle of 60 degrees measure a square. I not only say there is 
but one figure that will measure a circle, but I also have but 
one number that will measure a circle ; and that is the number 
6. With these — the hexagon and the number 6 — I have 
perfected the measure of the circle. 

If you take a trigon, of 3 sides, or a tetragon, of 4 sides, or 

a pentagon, of 5 sides, or a heptagon, of 7 sides, or an octagon, 

of 8 sides, or a nonagon, of 9 sides, or a decagon, of 10 sides, 

or an endecagon, of 11 sides, or a dodecagon, of 12 sides, either 

8 



86 t THE MEASURE OF THE CIRCLE. 

of these polygons would, in the operation, produce the surd 
number, and that would produce fractions, which I think no 
man would comprehend, as it would lead to the same difficulty 
which has been felt in all ages. 

Time is prefigurative of the number 6 ; and as time is meas- 
ured by a circle, I take the number 6 to measure the circle ; 
and the only polygon that is synonymous with that number is 
the hexagon. I am of the opinion that there is no other num- 
ber but the number 6 whereby the measure of the circle could 
have been effected. I should not have taken the number 6 
from choice, but from necessity, for 10 would have been prefer- 
able ; but 6 was the only number whereby I could measure it 
without surd numbers ; and I should think no man would pre- 
fer irrational numbers to work out a difficult problem, when he 
could have rational ones. The hexagon, or polygon of 6 sides, 
is the only one whereby I could get every angle, every side, 
and every measure perfect ; and if the foundation is imperfect, 
so would be the superstructure. 

Now, 6 X 6 = 36, equal the 2d power ; and 36 X 36 = 
1296, or the 4th power, which is the square of the square; 
and I can make nothing of the square of the square but the 
circle. 

The same question might be asked, why will not a tetrahe- 
dron, of 4 sides, or an octahedron, of 8 sides, or a dodecahe- 
dron, of 12 sides, or an icosahedron, of 20 sides? If any of 
these angles would have measured the circle, it is more than 
probable it would have been done years ago. 

THE NUMBER SIX. 

As I have before said, for my prime number in the measure 
of the circle I have taken the number 6. For this I refer to 
Plato, who says time is prefigurative of the number 6. From 
this observation of Plato's, our ancestors have thought the 
world was to last but 6000 years. In the 3d chapter of the 2d 
epistle of Peter the same opinion is confirmed. You there read 
of the scoffers and unbelievers ; and St. Peter's answer was, 



THE MEASURE OF THE CIRCLE. 87 

" Be not ignorant of this one thing ; that one day is with the 
Lord as a thousand years ; for slackness is not counted to the 
Lord as it is to man." St. Peter says it certainly will come. 
Now, as the world was 6 days in making, I draw the inference 
that it will stand 6000 years. This was the opinion of B. P. 
Burket, and many of our ancestors who were considered wise 
and great men. 

You may observe my ratio to measure a circle (which is a 
true figure of eternity) would naturally end with a 6; and 
thus you see a true ratio never could have been found by the 
natural use of mathematics. In the Revelation of St. John 
you find that after the 7th seal the world is measured to its end 
by 6 trumpets. Now, under 6 trumpets you find the four 
angels loosed from the great river Euphrates, by whose army 
the greater part of men were slain. This army was computed 
at 200,000,000. The oath in chapter 10 was, that when the 
7th trumpet began to sound, the end of the world should come ; 
so you see it lasts but through the 6. And in the 13th chapter 
you find, under the reign of the great beast, that no one had 
liberty to buy or sell except he had the mark of the beast, or 
the number of his name. Let him that hath understanding 
count his number ; and his number is 666. Now, it goes by 
thirds, as you may see by reading ; as 3 times 3 are 9, and 666 
multiplied by 9 is to the square equal to 5994. This makes it 
lack 6 of making the 6000 years. In the 24th chapter of 
Matthew our Savior says, " Except these days be shortened, 
no flesh shall be saved." Now, the number of the beast falls 
short of 6000 by 6 years ; so it appears that the time was short- 
ened 6 years ; and you see that 6 is the number that governs 
all these operations ; and it is evident that great operations can 
be performed with it. 

Now, I offer as proof of the number 6 : The number 6 is not 
proficient of the measure of the circle, therefore I go to the 
number 7 ; and what I take of the number 7 is synonymous 
with the lack of the 6000 years, which is my biquadrate, and 
is what makes up the measure of the circle. 



88 



THE MEASURE OF THE CIRCLE. 



Now, the hexagon has 6 sides, 6 squares, 6 angles, and 6 
radii ; the 6 radii are each 6 inches in length, the 6 squares 
each 6 inches in length, and the 6 angles each 6 inches in 
length. The 6 sides, each 6 inches, the 6 oblong squares, with 
the biquadrate added, is the perfect measure ; as 6 times 6 are 
36, and £ of 6 added to 6 times 6 is 38, which is the complete 
and scientific mathematical measure of the circle. 

Time was to be continued through the sounding of the 6 
trumpets ; and when the 7th began to sound, time was to be 
no more. Now the want of the measure of the circle is no 
more when I take of the number 7 my biquadrate. As I 
believe the measure of the circle was not to be effected until 
the sounding, or fulfilment of the Scriptures, and as it is gov- 
erned by the number 6, it was not to be measured in the sound- 
ing of the 1st, 2d, 3d, 4th, or 5th trumpet, but in the sound- 
ing of the 6th. 

TRIGONOMETRY. 

I am told that trigonometry has found the measure of the 
circle to a hair's breadth. What is meant by it I do not know, 
nor can I conceive. Trigonometry cannot compute the true 
perimeter of a circle, unless that circle be already known. 

Most men are aware that computation is governed by a circle ; 
they also believe — for they find it in practice — that there is a 
truer measure than the one now in use ; and if all measure- 
ment is governed by a circle, it comprehends a gallon as well 
as a bushel, and a right line as well as a curve ; and if the circle 
which directs these measures be incorrect, it is easy to see why 
all offsprings of that circle should be deficient in the same pro- 
portion. If we wish to compute a segment of a circle, we 
take spherical trigonometry to do so ; but we must understand 
at the same time that trigonometry cannot give us the truth, 
since we know its foundation is affected more or less by an 
error known by mathematicians as most dangerous for its 
counterbalance, wherein it contracts truth, shows truth, proves 
truth, and is still a great falsehood. We must therefore be 



THE MEASURE OF THE CIRCLE. 89 

acquainted with the true circle before we can find its true parts ; 
and as trigonometry is but that portion of science which exe- 
cutes the laws of geometry, and as geometry has never declared 
a circle to be equal to any figure but itself, we cannot take trig- 
onometry to give us a segment, or a whole circle, within a hair's 
breadth, or any other breadth, with confidence. 

I am told that the pendulum swings a perfect yard. This I 
do not understand. It must take a very ingenious mechanic to 
make a pendulum swing a perfect yard, when no man as yet 
but myself ever knew what length a perfect yard was. 

If we admit that the length of a yard was known to man, it 
would be impossible, from the nature of things, to make a pen- 
dulum that would swing a perfect yard, because the influence 
of the atmosphere upon the metal would cause it to vary more 
from correct measure than as 7 to 22 does. It has not as yet 
been ascertained that any composition of metals can be so 
worked as to obviate the variation caused by atmospheric influ- 
ence ; so that this doctrine is much like that of the gentleman 
professor in England who told me that if I worked a problem 
right to measure a circle, there would always be a remainder ; 
therefore, in order to do a thing right, I must do it wrong ; for 
my work was not mathematical because it came right. This is 
a strange doctrine, although it is believed by a great many in 
this world. 

QUESTIONS ASKED. 

I am told by the learned classes that the circle is measured 
.as near as it ever can be, and near enough for any thing. I 
will ask a number of questions, which I think will illustrate the 
work to the scholar : — 

1. From what do all measures spring, or from what derived? 

2. From what do all weights spring, or from what derived ? 

3. What makes a mathematical inch ? how do you arrive at 
this? 

4. What makes a pound or an ounce? how r II? 



90 THE MEASURE OF THE CIRCLE. 

5. What is the measure of the circle, and what is the use 
of it? 

6. Why has it remained in oblivion if it is useful to man ? 

7. How do you find the circumference of a circle ? 

8. How do you find the area of a circle ? 

9. Having the circumference, how do you find the diameter ? 

10. Having a promiscuous number of figures for an area, 
how do you find the circle that bounds them ? 

11. What is the 4th power, or biquadrate ? 

12. What part of a circle is the biquadrate ? 

18. What are the square inches of the dodecagon ? 

14. What makes up the measure of the circle ? 

15. What is the square of the circle ? 

16. What will be the size of a circle from the square of 9^ 
inches ? 

17. What will be the size of the square from a circle 12 
inches in diameter ? 

18. What is the difference between square and circular 
measure ? 

19. What angle is square measure from? 

20. What angle is circular measure from ? 

21. What is the difference between the two angles ? 

22. What measure is that which gains over all others ? 

23. What proportion is the square to the circle ? 

24. How much does the circle gain over the square ? 

25. How do you measure a cone ? 

26. How do you measure a cylinder ? 

27. How do you find a level on the earth ? 

28. How do you find the curve of the earth. ? 

29. What is the curve of the globe ? 

30. What proportion has the diameter of a circle to its cir- 
cumference ? 

31. What is the difference between as 7 to 22 and as 6 to 
19? 

32. How much is long measure astray ? 

33. How much is square measure astray ? 



THE MEASURE OF THE CIRCLE. 91 

34. How much is cubic measure astray ? 

35. How much do 1728 cubic inches of water weigh ? 

36. How many cubic inches are there in a bushel ? 

37. What portion of 1728 cubic inches is one ounce ? 

38. What portion of 1728 cubic inches is one pound? 

39. What is time measured by ? 

40. Why does the number 6 measure time? 

41. Why does the number 6 measure a circle? 

42. Why is the hexagon the only figure or angle to measure 
a circle ? 

43. What is a surd number ? 

44. How do you extract the square root of the surd number ? 

45. What makes a surd number ? 

46. What portion of a cylinder, supposing it to be square, 
to find the measure in square inches, is the circle of it ? 

47. How do you find the square inches in a cylinder, and by 
what measure ? 

48. How do you measure a grindstone ? 

49. How do you measure land, if in a circular form ? 

50. By what rule do you find the square inches in a steam 
boiler ? 

51. By what rule do you find the square inches in a globe? 

52. By what rule do you measure a cask correctly? 

53. Can square measure perfect circular? 

54. Can circular measure perfect square measure ? 

55. What measure has been sought for in all ages of the 
world ? 

56. What measure was perfected in 1845 ? 

57. What does the angle gain on the circle? 

58. What is three times the square of the radius in a circle 
12 inches in diameter ? 

59. Why does the circle measure the area ? 

60. Why does the area measure the circle ? 

61. Why does the square prove the circle? 

62. Why dees the circle prove the square ? 

63. Why does the square prove the area? 



92 THE MEASURE OF THE CIRCLE. 

64. Why does the area prove the square ? 

65. Suppose you form a circle from a piece of wire 38 inches 
long, and a square from a piece 38 inches long ; what is the 
difference of the area and the square ? 

66. Why is one half the diameter one fourth in area? 

67. What is the square root of a circle 12 inches in diam- 
eter ? 

68. What is the difference between the square of the circle 
and the square of the area ? 

69. What square will a circle form that is 12 inches in 
diameter ? 

70. What circle will a square make that is 9^ inches square ? 

71. What circle is that which has 112 for area? 

72. What area is that which has a circle 12 inches in diam- 
eter ? 

73. What area has a circle 17 inches in diameter ? 

74. What area has a circle that has a radius of 16 inches ? 

75. What would be the radius of a circle that has 224 inches 
area? 

76. What is the difference of area between a circle 12 inches 
in diameter, and one 6 inches in diameter? 

77. What is the difference in the length of a string that 
goes round the circle, and one that goes round the square ? 

78. How do you find the difference of area between a circle 
and a square ? 

79. What is the difference of area of any circle derived from 
as 7 to 22, or as 6 to J.9? 

80. Suppose you travel on a radius 100 miles ; what is the 
difference of circular measure from as 6 to 19 or as 7 to 22 ! 

81. Suppose you shot a ball level on the earth 1000 miles, 
and it then fell perpendicularly to the earth ; how far would it 
have to fall ? 

82. Suppose your radius is 100 miles ; how much larger is 
your circle ? 

83. Suppose your radius is 40 miles; what is the area of 
the circle? 



THE MEASURE OF THE CIRCLE. 9<3 

84. Suppose your circle is 4 times the radius ; what is the 
area? 

85. Suppose one third of your radius is 12 inches ; what is 
the circumference of your circle ? 

86. What is the biquadrate when your radius is 12 inches ? 

87. What is the biquadrate when the radius is 34- inches ? , 

88. What is the biquadrate when your circle is 78 inches ? 
^ 89. What is the biquadrate when the area is 240 inches ? 

90. What is the biquadrate of a circle which has an area of 

17 inches ? 

MEASURES FOR GRAIN, ETC. 

1. A measure to hold a bushel, if in circular form, must be 

18 inches in diameter, and 9 inches, lacking tVtt of an inch, in 
depth. 

2. A measure to hold half a bushel, if in circular form, must 
be 12£ inches in diameter, and 9 inches deep. 

3. A measure to hold a peck, if in circular form, must be 10 
inches in diameter, and 7£ inches deep. 

4. A measure to hold 4 quarts, if in circular form, must be 
8£ inches, lacking .019f, in diameter, and must be 5 inches 
deep, which is a little too much, as 280 to 282. 

5. A measure to hold 2 quarts, if in circular form, must be 
6 inches in diameter, and 5^V inches deep. 

6. A measure to hold 1 quart, if in circular form, must be 
4£ inches in diameter, and 5 inches in depth. 

7. A measure to hold a pint, if in circular form, must be 3 
inches in diameter, and 5 inches deep. 

8. A measure to hold half a pint, if in circular form, must 
be 2£ inches in diameter, and 3 T 6 <j inches deep. 

9. A measure to hold a gill, if in circular form, must be 2 
inches in diameter, and 3 inches deep. 

10. A measure to hold a glass, if in circular form, must be 
l£ in diameter, and 2% inches deep. 

On the following page is a table showing what length to cut 
iron that is used for hoops, wagon tires, and all such uses as 



94 



THE MEASURE OF THE CIRCLE. 



require a length to be cut for a circumference. Nothing is 
allowed for drawing ; I allow the iron to be half an inch thick. 
I calculate from a diameter of 6 feet to one of 1 foot : — 



Length. 


Circumference. 


Length of hoop. 


Length. 


Circumference. 


Length of hoop. 


Feet. In. 


Feet. 


Inches. 


Feet 


Inches. 


Feet. 


In. 


Feet 


Inches. 


Feet. 


Inches. 


6 


19 




18 


1H 


3 


5 


10 


H 


10 


lOf 


5 11 


18 


8.8$ 


18 


B| 


3 


4 


10 


6* 


10 


74 A 


5 10 


18 


5.4| 


18 


HA 


3 


3 


10 


3f 


10 


4fA 


5 9 


18 


n 


18 


HA 


3 


2 


10 


Of 


10 


14 


5 8 


17 


Hi 


17 


10| 


3 


1 


9 


9* 


9 


104 A 


5 7 


17 


84 


17 


7fA 


3 





9 


6 


9 


7 2 

' TF 


5 6 


17 


5 


17 


4|t 2 f 


2 


11 


9 


n 


9 


4 


5 5 


17 


1* 


17 


1| 


2 


10 


8 


ii* 


9 


Of A 


5 4 


16 


10* 


16 


10* A 


2 


9 


8 


8f 


8 


94 A 


5 3 


16 


7| 


16 


7 2 


2 


8 


8 


•I 


8 


64 


5 2 


16 


>f 


16 


4 


2 


7 


8 


2* 


8 


Q2 1 
°6" T¥ 


5 1 


16 


1* 


16 


tA 


2 


6 


7 


11 


8 


of A 


5 


15 


10 


15 


9* A 


2 


5 


7 


H 


7 


9| 


4 11 


15 


H 


15 


6* 


2 


4 


7 


H 


7 


74 A 


4 10 


15 


H 


15 


34 A 


2 


3 


7 


n 


7 


4 -3r 


4 9 


15 


Of 


15 


Of A 


2 


2 


6 


n§ 


7 


1 


4 8 


14 


91 


14 


H 


2 


1 


6 


8* 


6 


9fA 


4 7 


14 


H 


14 


64 


2 





6 


4 


6 


64 A 


4 6 


14 


3 


14 


Q 2 
IF 


1 


11 


6 


06 


6 


24 


4 5 


13 


Hf 


14 





1 


10 


5 


9-1 


5 


UfA 


4 4 


13 


8* 


13 


8*tV 


1 


9 


5 


H 


5 


«2 _2 


4 3 


13 


«f 


13 


HA 


1 


8 


5 


H 


5 


5 F 


4 2 


13 


2f 


13 


24 


1 


7 


5 


o* 


5 


24 A 


4 1 


12 


11* 


12 


ii* A 


1 


6 


4 


9 


4 


11 A 


4 


12 


8 


12 


8f 


1 


5 


4 


5* 


4 


8 


3 11 


12 


H 


12 


*t 


1 


4 


4 


n 


4 


44 


3 10 


12 


i* 


12 


24 A 


1 


3 


3 


hi 


4 


if A 


3 9 


11 


10f 


11 


ii A 


1 


2 


3 


H 


3 


10| 


3 8 


11 


71 


11 


74 


1 


1 


3 


«i 


3 


7fA 


3 7 


11 


*4 


11 


44 A 


1 





3 


2 


3 


4|A 


3 6 


11 
- — 


If 


10 


ii* A 















To find the height of a steeple : Take a quadrant, and place 
yourself a distance of 45 degrees from the base ; that is, place 
yourself at such a distance as to make the quadrant strike the 
top of the steeple ; then measure from the base to where you 
stand, and allow for your height, and that will be the height of 
the steeple. 



THE MEASURE OF THE CIRCLE. 95 

To show the loss on linear, square, and cubic measure : 

2)132(66 3)132(44 

If 132 : 1 : : 100 If 66 : 1 :: 100 If 44 : 1 : : 100 
100 100 100 



132) 1000 (.76 per cent. 66)100(1.51 44)100(2.28 
924 66 88 

76 340 120 

330 88 

100 320 

66 352 

34 little short. 

So it makes linear measure near or about 76 per cent. ; square 
measure, about 1.51 per cent. ; and cubic measure about 2.28 
per cent. But as you cannot use the angle in the work, it gives 
too little in cubic measure, and weights are in proportion of 
that measure. Now, the statute of America says that a gallon 
of water weighs just 8 pounds, and 221 inches make a gallon, 
and 1 cubic foot of water weighs 62^- pounds ; so, as measure- 
ments of all kinds are wrong, so are weights ; and weights are 
wrong in proportion of cubic measure. 

In England, the wine gallon is 231 inches, which is 10 
inches more than it is in America ; and yet they reckon 8 
pounds to the gallon, which shows their weights and measures 
to be about ^V part more than the American weights and meas- 
ures, which accounts for the remark common in England when 
I was a boy, that when any thing was short of measure it was 
Presbyterian measure ; the reason for which was, I presume, 
that the Presbyterians came to America to enjoy their religious 
freedom, and made their measures a little less ; so the English 
called it Presbyterian measure. The foot and inch, however, 



96 THE MEASURE OF THE CIRCLE. 

are the same in both, countries. The English call 231 inches a 
gallon, and the Americans 221; but the right measure would 
be 226 ; so the English gallon is 5 inches too much, and the 
American 5 too little ; all weights, therefore, should be altered 
in the same proportion. 

NUMERATION. • 

It is seldom that persons numerate more than 12 figures. I 
saw a plan in New Haven, Connecticut, which was new to me ; 
it ran to 10 figures, for millions, and after that every denomina- 
tion went by 3 figures, thus : Billions, tens of billions, hun- 
dreds of billions ; trillions, tens of trillions, hundreds of tril- 
lions, &c. 

But I think the way is to numerate by logarithms, thus : 3 
figures for hundreds, 6 for thousands, 12 for millions, 24 for 
billions, 48 for trillions, 96 for quadrillions, &c. ; there would 
then be no number that could not be numerated ; if there were 
100 figures I could numerate them, but I never found another 
that could. 

PERMUTATION. 

The English language has 20,500 nouns, 40 pronouns, 9,500 
adjectives, 8,000 verbs, 2,600 adverbs, 69 prepositions, 19 
conjunctions, 68 interjections, and 2 articles ; making in all 
40,000 words in the language. 

The alphabet has 26 letters. I have worked to find how 
many different changes there are in the English alphabet, and 
£nd them to be 4,226,579,232,623,185,627,342,000,000. I 
numerate this thus : 4226 trillions, 579,232 millions of bil- 
lions, 623,185 billions, 627,342 millions. 

These are the different ways that the 26 letters are changed 
to make words and representations. In this numeration there 
are two ciphers too many. 



THE MEASURE OF THE CIRCLE. 97 



NUMERATION TABLE. 



O 

o 

o 

la 

o 
o 
o 

»— » 
Or 
\* 

SD 
t— > 
-J 

V. 

Or 
^ 

OS 
t— ' 

*© 

I— 1 

CD 
QC 

«• 
-3 

O 
Cr» 

to 
I— * 

J® 



Units. 
Tens. 
Hundreds. 



Thousands. 

Tens of thousands. 

Hundreds of thousands. 



Millions. 

Tens of millions. 

Hundreds of millions. 



Thousands of millions. 

Tens of thousands of millions. 

Hundreds of thousands of millions. 



Billions. 

Tens of billions. 

Hundreds of billions. 



Thousands of billions. 

Tens of thousands of billions. 

Hundreds of thousands of billions. 



Millions of billions. 

Tens of millions of billions. 

Hundreds of millions of billions. 



Thousands of millions of billions. 

Tens of thousands of millions of billions. 

Hundreds of thousands of millions of billions. 



Trillions. 

Tens of trillions. 

Hundreds of trillions. 



Thousands of trillions. 

Tens of thousands of trillions. 

Hundreds of thousands of trillions. 



Millions of trillions. 

Tens of millions of trillions. 

Hundreds of millions of trillions. 



Thousands of millions of trillions. 

Tens of thousands of millions of trillions. 

Hundreds of thousands of millions of trillions. 



Billions of trillions. 

Tens of billions of trillions. 

Hundreds of billions of trillions. 



Thousands of billions of trillions. 

Tens of thousands of billions of trillions. 

Hundreds of thousands of billions of trillions. 



Millions of billions of trillions. 

Tens of millions of billions of trillions. 

Hundreds of millions of billions of trillions. 



Thousands of millions of billions of trillions. 

Tens of thousands of millions of billions of trillions. 

Hundreds of thousands of millions of billions of trillions 



98 THE MEASURE OF THE CIRCLE. 

How many changes can be rung on 12 bells ? 



1 


1 




2 


2 


2 




3 


3 


6 




4 


4 


24 




5 


5 


120 




6 


6 


720 




7 


7 


5040 




8 


8 


40320 




9 


9 


362880 




10 


10 


3628800 



11 



3628800 
3628800 



11 39916800 
12 



79833600 
39916800 



12 479001600 



THE MEASURE OF THE CIRCLE. 99 

MULTIPLICATION 

Is an expeditious way of performing several additions, as in 
this case: Required to multiply £ 1 19s. 11 d. 3 qr. by itself; 
what will be the answer ? Ans. £3 19 s. lid. 2 qr. 

THE MILE IN DIFFERENT COUNTRIES. 

The English mile contains 1760 yards. 

The Russian mile contains 1100 yards. 
The Irish and Scotch mile contains 2200 yards. 

The Staton mile contains 1460 yards. 

The Polish mile contains 4400 yards. 

The Spanish mile contains 5028 yards. 

The German mile contains 5866 yards. 

The French league contains 3666 yards. 

Algebra is the art and science of some ingenious, cunning 
thoughts, calculated to fill the scholar's mind, and take his 
labors for foolish employment, which will do him no good, but 
darken his literary prospects. I would advise all to have noth- 
ing to do with it, as I consider it useless to all science and lit- 
erature. It is a labored, worthless study, calculated by craft, 
for the sake of fame. It gives the scholar much labor and 
anxiety, which keeps his mind from other useful and important 
sciences. 

Doctrine of annuities is to find the value of any annuity. 

Logarithms are certain artificial numbers. 

Trigonometry is finding all the sides and angles of a triangle. 

Astronomy is the science of the heavenly bodies. 

Mechanical powers are six in number. 

Electricity is combustible air, such as lightning. 

Medical electricity is applying it to different uses. 

Pneumatics is that part of natural philosophy that treats of 
the weight, pressure, and electricity of air. 



100 THE MEASURE OF THE CIRCLE. 

Hydrostatics treats of the equilibrium of fluids, or the gravi- 
tation of fluid bodies. 

All fluids except air are incompressible. 

Fluidity, according to Sir Isaac Newton's definition, is the 
yielding of a body when compressed by any force, the parts of 
the body being easily moved. 

A quart of air weighs 16 grains. 

Hydraulics is the science of the force and motion of fluids. 

Geometrical progression is when the numbers increase thus : 
1, 2, 4, 8, 16, 32, 64, 128, &c. 

Arithmetical progression is when the numbers increase thus : 
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39. 

The number of inches in an English gallon is, for oil, 282, 
for wine, 231. In America it is 221 inches. In a bushel 
there are 2256 inches; in half a bushel, 1128; in one peck, 
564 ; in one gallon, or half a peck, 282 ; in one quart, 70^ ; 
in one pint, 35j ; in half a pint, 17 T 6 ^ and £ ; in a gill, 8j- ; in 
a glass, 4£ ; in half a glass, 2J. 

Wightman, in his arithmetic, says there are 2218 inches in a 
bushel, and he also says it measures in diameter 19£ inches, and 
is 18 £ inches deep. Now, according to this he would have 
2441 inches in a bushel. 

Thomas Hodson says, in his " Tutor," that there are 282 
inches in an oil gallon, and he makes a bushel to contain 2256 
inches, that is, if you take the oil measure ; and if we take our 
statute gallon of 221 inches, the bushel will be 1768 inches. 

Now, which of these is right is not for me to say ; I leave 
that for those who may be interested. 

Euclid, the celebrated mathematician, lived 277 years before 
Christ. 

Archimedes lived 208 years before Christ. 
Ptolemy lived 148 years before Christ. 



THE MEASURE OF THE CIRCLE. 101 

John Napier, of Scotland, lived in 1600, and discovered 
logarithms. 

James Gregory lived in 1675. He was also of Scotland. 

Isaac Barron lived in 1677. He was also of Scotland. 

John Flamsteed lived in 1719. He was of Derbyshire, in 
England. 

Dr. John Kell lived in 1719. 

Sir Isaac Newton lived in 1700. He was of Lincolnshire. 

The above were celebrated mathematicians. 

Pythagoras, one of the greatest philosophers, considered 
mathematics of great importance to man. He says mathemat- 
ics is the first step towards wisdom. Lord Bacon says men do 
not sufficiently understand the value and importance of mathe- 
matics. It is a noble and divine system, calculated for the ben- 
efit and enterprise of mankind generally. 

GEOMETRY. 

There are three different angles — the right angle, the obtuse 
angle, and the acute angle. 

When a plain figure is terminated by three lines it is called 
a triangle. 

There are three particular sorts of triangles — equilateral, 
isosceles, and right angled. 

When the three sides of a triangle are equal, it is called an 
equilateral triangle. 

When two sides only are equal it is called an isosceles tri- 
angle. 

When one angle is a right angle, it is called a right angled 
triangle. 

Any side of a triangle may be called the base, and the angle 
opposite to it is called the vertex. 

The side opposite to a right angle is called the hypotenuse 
9* 



102 THE MEASURE OF THE CIRCLE. 



LINEAR MEASURE. 

12 inches make 1 foot. 
3 feet " 1 yard. 

6 feet " 1 fathom. 

16 J feet, or 5£ yards, make 1 pole or rod. 
40 poles make 1 furlong. 
8 furlongs " 1 mile. 



SQUARE MEASURE. 

144 inches make 1 foot. 

9 feet " 1 yard. 

36 feet " 1 fathom. 

272^ feet, or 30J yards, make 1 rod. 

1600 poles make 1 furlong. 

64 furlongs " 1 mile. 



160 rods make 1 acre. 
80 rods equal } an acre. 
40 rods i( i of an acre. 
20 rods " £ of an acre. 
10 rods " T V of an acre. 
5 rods t( -3V of an acre. 
640 acres make 1 square mile. 
144 square inches make 1 square foot. 
1728 cubic inches make 1 cubic foot. 

On the opposite page will be found the compound interest 
of one dollar, from 1 year to 12, at 6 per cent. 

One dollar, at 6 per cent., will double in 12 years ; and the 
simple interest of one dollar for 12 years would be 72 cents; 
so the compound interest of one dollar for 12 years is 28 
cents. 






THE MEASURE OF THE CIRCLE. 



103 



The interest of 
one dollar for 12 
years, at simple 
interest, -would be 
72 cents. 



1.00 
6 



6.00 
100 



106.00 
12 



6, 
100 



1.06 
6 



#1.7200 
Compound, 28 

#2.00 



6.36 
106 

1.1236, 
6 

6.7416 
112.36 



lyear. 



2 years. 



1.191016, 3 years. 
6 



7.146096 
1.191016 



1.26247696, 4 years. 
6 



7.57486176 
126.247696 



1.3382255776, 5 years. 



8.0293534656 
1.3382255776 



1.418519112256, 6 years. 
6 



8.511114673536 
1.418519112256 



1.50363025899136, 7 years 
6 



9.02178155394816 
1.50363025899136 



1.5938480745308416, 8 years. 
6 



9.5630884471850496 
1.5938480745308416 



9 years. 



10.130873753416152576 
1.688478958902692096 

1.78978769643685362176, 10 years. 
6 

10.73872617862112173076 
1.78978769643685362 176 • 

1.8871749582230648390676, 11 years. 
6 

11.3230497493383890344056 
1.8871749582230648390676 

2.000405455716448729411656, 12 years. 



104 



THE MEASURE OF THE CIRCLE. 



The manifestation of the spirit is given to every man, to 
profit withal. To one is given wisdom ; to another is given 
knowledge ; to another is given faith ; to another, the gift of 
healing ; to another, the working of miracles ; to another, 
prophecy ; to another, discerning of spirits ; to another, divers 
kinds of tongues ; to another, interpretation of tongues. 

But all these work that selfsame spirit, dividing to every 
man severally as he will. 

A TABLE OF GAUGING. 

I find the mean diameter by adding the bung and head 
together ; one half of this will be the mean diameter. Multi- 
ply by the length, which gives the contents, in cubic inches ; 
divide by 221 inches, the number of inches in a gallon, as this 
table is calculated, which gives the gallons. The table is laid 
down in columns every inch in length, with decimals if the 
length is short or long, for the diameter laid down ; one state- 
ment in the rule of three will find it, or you may find how 
much it is too long or short by tabular measure. Suppose it 
is 30 inches long in the table, and cask 32 ; then your cask 
will hold 15 gallons more. 



Mean 






Mean 






Mean 




! 


diameter. 


Length. 


Gallons 


diameter. 


Length. 


Gallons. 


diameter. 


Length. 

44 


Gallons. | 


39.9 


48 


269.7 


38.4 


46 


238.8 


36.9 


210.95 


39.8 


48 


267.7 


38.3 


46 


237.5 


36.8 


44 


209.8 


39.7 


48 


266.38 


38.2 


46 


236 


36.7 


44 


208.67 


39.6 


48 


265.04 


38.1 


46 


235 


36.6 


44 


207.35 


39.5 


48 


263.7 


38 


46 


233.8 


36.5 


44 


206.4 


39.4 


47 


256.9 


37.9 
37 8 


46 
45 
45 


232.65 

226.4 

225.2 


36.4 


44 


205.27 


39.3 


47 


255.6 


36.3 


44 


204.14 


39.2 


47 


254.25 


37.7 


36.2 


44 


203 


39.1 


47 


253 


37X 
37.5 
37.4 


45 
45 
45 


224 

222.8 

221.6 


36.1 


44 


201.9 


39 


47 


251.7 


36 

~35.9 


44 
~43~ 


200.7 


38.9 


47 


250.42 


195 


38.8 


46 


243.8 


37.3 


45 


220.5 


35.8 


43 


194 


38.7 


46 


242.5 


37.2 


45 


219.7 


35.7 


43 


192.9 


38.6 


46 


241.3 


37.1 


45 


218.09 


35.6 


43 


191.9 


38.5 


46 


240.08 


37 


45 


216.9 


35.5 


43 


190.8 



THE MEASURE OF THE CIRCLE. 



105 



TABLE OF GAUGING, CONTINUED. 



| Mean 
•diameter. 






1 Mean 






Mean 






Length. 

43 


Gallons. 

189.7 


diameter. 


Length. 

37 


Gallons. 


diameter. 


Length. 

34 


Gallons. 

86.5 


35.4 


31.1 


126 


26.9 


35.3 
35.2 


43 
43 


188.66 
187.6 


31 

30.9 


37 


125.2 
T21 


26.8 
26.7 


34 
34 


85.9 
85.3 


36 


35.1 


43 


186.5 


30.8 


36 


120.25 


26.6 


34 


84.7 


35 


43 


185.4 

180.2 


30.7 
30.6 


36 
36 


119.7 
118.66 


26.5 
26.4 


34 
34 


84 
83.29 


34.9 


42 


34.8 


42 


179.3 


30.5 


36 


117.9 


26.3 


34 


82.8 


34.7 


42 


178.4 


30.4 


36 


117.1 


26.2 


34 


82.14 


34.6 


42 


177.5 


30.3 


36 


116.3 


26.1 


34 


81.55 


34.5 
34.4 


42 
41 


176.6 
170.7 


30.2 
30.1 


36 
36 


115.6 
114.49 


26 


34 


80.9 


25.9 


33 


77.94 


34.3 


41 


169.85 


30 


36 


114 


25.8 


33 


77.33 


34.2 


41 


168.29 


29.9 


36 


113.29 


25.7 


33 


76.74 


34.1 


41 


167.87 


29.8 


36 


112.26 


25.6 


33 


75.79 


34 


41 


166.8 


29.7 


36 


111.8 


25.5 


33 


75.5 


33I" 


40 


161.85 


29.6 


36 


111 


25.4 


33 


74.9 


33.8 


40 


160.95 


29.5 


36 


110.3 


25.3 


33 


74.39 


33.7 


40 


159.92 


29.4 


36 


109.56 


25.2 


33 


73.79 


33.6 


40 


159 


29.3 


36 


108.8 


25.1 


33 


73.2 


33.5 
33.4 


40 
40 


158.6 
157.8 


29.2 
29.1 


36 
36 


107.7 
107.3 


25 


33 
~32~ 


72.6 
69.85~ 


24.9 


33.3 


40 


156.88 


29 


36 


106.6 


24.8 


32 


69.18 


33.2 


40 


155.24 


28.9 


35 


102.9 


24.7 


32 


68.74 


33.1 


40 


154.31 


28.8 


35 


102.2 


24.6 


32 


68.19 


33 


40 


153.37 
Hs.64 


28.7 
28.6 


35 
35 


101.5 

100.8 


24.5 
24.4 


32 
32 


67.59 
67.04 


32.9 


39 


32.8 


39 


147.9 


28.5 


35 


100.1 


24.3 


32 


66.5 


32.7 


39 


146.78 


28.4 


35 


99.4 


24.2 


32 


65.95 


32.6 


39 


145.94 


28.3 


35 


98.7 


24.1 
24 


32 


6?.4 


32.5 


39 


145 


28.2 


35 


98 


32 


64.9 


32.4 


39 


144.15! 


28.1 


35 


97.3 


~2~K9~ 


~31~ 


62.3 


32.3 


39 


143.26 


28 


35 


96.6 


23.8 


31 


61.82 


32.2 


39 


142.39 


27.9 


35 


~ 96.2 


23.7 


31 


61.3 


32.1 


39 


141.5 


27.8 


35 


95.24 


23.6 


31 


60.79 


32 


39 
38 


140.6 
136.15 


27.7 

27.6' 


35 
35 


94.5 

93.87 


23.5 
23.4 


31 
31 


60.27 
59.76 


31.9 


31.8 


38 


135.3 


27.5 


35 


93.5 


23.3 


31 


59.72 


31.7 


38 


134.4 


27.4 


35 


92.64 


23.2 


31 


58.71 


31.6 


38 


133.6 


27.3 


35 


91.8 


23.1 


31 


58.2 


31.5 
31.4 


38 
37 


132.75! 
128.4 


27.2 
27.1 


35 
35 


91.17 

90.5 


23 


31 

30 


57.7 
55.36 


22.9 


31.3 


37 
37 


127.6 ! 
126.8 | 


27 

_ 


35 


89.84 


22.8 
22.7 


30 


54.9 


31.2 






1 


30 


54.4 



106 



THE MEASURE OF THE CIRCLE. 



TABLE OF GAUGING, CONTINUED. 



Mean 






Mean 






Mean 


i 




diameter. 


Length. 


Gallons. 


diameter. 


Length. 

28 


Gallons. 


diameter. 


Length. 


Gallons. 


22.6 


30 


53.95 


19.4 


37.1 


15.1 


24 


19.26 


22.5 


30 


53.47 


19.3 


28 


36.72 


15 


24 


19 


22.4 


30 


53 


19.2 


28 • 


36.34 


14.9 
14.8 
14.7 
14.6 


23 

23 
23 
23 


17.97 
17.73 
17.5 


22.3 


30 


52.5 


19.1 


28 


35.26 


22.2 
22.1 


30 
30 


52.1 
51.5 


19 


28 


35.23 


18.9 


27 


33.96 


17.26 


22 


30 


51.1 

48.94 


18.8 
18.7 


27 

27 


33.6 
33.2 


14.5 
14.4 


23 
23 


17 


21.9 


29 


17.13 


21.8 


29 


48.52 


18.6 


27 


32.82 


14.3 


23 


16.54 


21.7 


29 


48.2 


18.5 


27 


32.5 


14.2 


23 


16.32 


21.6 


29 


47.6 


18.4 


27 


32.17 


14.1 


23 


16.1 


21.5 


29 


47.2 


18.3 


27 


31.8 


14 


23 


15.87 


21.4 
21.3 
21.2 
21.1 


29 
29 

29 
29 


46.7 
46.3 
46.89 
45.46 


18.2 
18.1 

18 


27 
27 
27 


31.4 

31.14 

30.8 

29.37 


13.9 
13.8 
13.7 
13.6 


22 
22 
22 
22 


14.96 
14.75 
14.53 


17.9 


26 


14.32 


21 


29 


45 


17.8 
17.7 


26 
26 


28.98 
28.67 


13.5 
13.4 


22 

22 


14.11 


20.9 


28 


43.6 


13.9 


20.8 


28 


42.65 


17.6 


26 


28.35 


13.3 


22 


13.77 


20.7 


28 


42.24 


17.5 


26 


28.03 


13.2 


22 


13.49 


20.6 


28 


41.83 


17.4 


26 


27.7 


13.1 


22 


13.29 


20.5 


28 


41.4 


17.3 


26 


27.4 


13 


22 


13.28 


20.4 


28 


41 


17.2 


26 


27.38 


12.9 


21 

21 


12.3 
12.11 


20.3 


28 


40.62 


17.1 


26 


26.76 


12.8 


20.2 


28 


40.2 


17 


26 


26.4 


12.7 


21 


31.92 


20.1 


28 


39.83 


16.9~ 


25 


125.14 


12.6 


21 


11.70 


20 


28 


39.46 


16.8 


25 


24.84 


12.5 


21 


11.55 


Barrels. 






16.7 


25 


24.5 


12.4 


21 


11.36 


20.9 


29 


44.6 


16.6 


25 


24.2 


12.3 


21 


11.18 


20.8 


29 


44.17 


16.5 


25 


23.95 


12.2 


21 


11 


20.7 


29 


43.75 


16.4 


25 


23.67 


12.1 


21 


10.82 


20.6 
20.5 


29 

29 


43.4 
42.9 


16.3 
16.2 


25 
25 


23.39 
23 


12 


21 


10.6 


11.9 


20 


9.97 


20.4 


29 


42.49 


16.1 


25 


22.8 


11.8 


20 


9.8 


20.3 
20.2 


29 
29 


42.07 
41.66 


16 


25 


22.5 
~21~.36 


11.7 
11.6 


20 
20 


9.64 


15.9 


24 


9.44 


20.1 


29 


41.25 


15.8 


24 


21.09 


11.5 


20 


9.31 


20 


29 
~~28~ 


40.8 
~39~~ 


15.7 
15.6 


24 
24 


20.87 
20.56 


11.4 
11.3 


20 

20 


9.15 


19.9 


8.99 


19.8 


28 


38.6 


15.5 


24 


20.2 


11.2 


20 


8.86 


19.7 


28 


38.26 


15.4 


24 


20 


11.1 


20 


8.67 


19.6 


28 


37.87 


15.3 


24 


19.78 
19.52 


11 


20 


8.52 


19.5 


28 


37.48 


15.2 


24 


1 







THE MEASURE OF THE CIRCLE. 



107 







TABLE 


OF GAUGING, CONTINUED. 






Mean 






Mean 






Mean 






diameter. 


Length. 


Gallons. 


diameter. 


Length. 


Gallons. 


diameter. 


Length. 

17 


Gallons. 


10.9 


19 


7.94 


9.9 


18 


6.21 


8.9 


4.74 


10.8 


19 


7.8 


9.8 


18 


6.08 


8.8 


17 


4.63 


10.7 


19 


7.65 


9.7 


•18 


5.95 


8.7 


17 


4.53- 


10.6 


19 


7.51 


9.6 


18 


5.84 


8.6 


17 


4.4 


10.5 


19 


7.37 


9.5 


18 


5.72 


8.5 


17 


4.32 


10.4 


19 


7.23 


9.4 


18 


5.6 


8.4 


17 


4.22 


10.3 


19 


7.9 


9.3 


18 


5.47 


8.3 


17 


4.12 


10.2 


19 


6.96 


9.2 


18 


5.35 


8.2 


17 


4.02 


10.1 


19 


6.82 


9.1 


18 


5.24 


8.1 


17 


3.82 


10 


19 


6.69 


9 


18 


5.13 


8 


1 17 


3.8 



JOHN Q. ADAMS S REPORT. 

I shall here introduce quotations from the report of Mr. 
Adams, Secretary of State, on weights and measures, ordered 
by the Senate of March 3, 1817, and presented February 22, 
1821. The report in full would occupy too much space, or I 
should be happy to present it ; but I shall give enough to show 
that he was in need of the measure of the circle to complete a 
useful report. Had Mr. Adams and Mr. Jefferson known the 
true measure of the circle, their labor to attain their object 
might be confined to personal judgment altogether, independent 
of former instructions. Mr. Adams says, — 



When weights and measures present themselves to the con- 
templation of the legislator, and call for the interposition of 
law, the first and most prominent idea which occurs to him is 
that of uniformity ; his first object is to embody them into a 
system, and his first wish to reduce them to one universal com- 
mon standard. His purposes are, uniformity, permanency, uni- 
versality, — one standard to be the same for all persons and 
all purposes, and to continue the same forever. These pur- 
poses, however, require powers which no legislator has hitherto 
been found to possess. The power of the legislator is limited 
by the extent of his territories and the numbers of his people. 
His principles' of universality, therefore, cannot be made by the 



108 THE MEASURE OF THE CIRCLE. 

mere agency of his power to extend beyond the inhabitants of 
his own possessions. The power of the legislator is limited 
over time ; he is liable to change his own purposes ; he is not 
infallible ; he is not immortal ; his successor accedes to his 
power, with different views, different opinions, and perhaps 
different principles. The legislator has no power over the prop- 
erties of matter ; he cannot give a new constitution to nature ; 
he cannot repeal her law of universal mutability ; he cannot 
square the circle ; he cannot reduce extension and gravity to 
one common measure ; he cannot divide or multiply the parts 
of the surface, the cube or the sphere, by the uniform and ex- 
clusive number 10. The power of the legislator is limited 
over the will and actions of his subjects. His conflict with 
them is desperate, when he counteracts their settled habits, 
their established usages, their domestic and individual economy, 
their ignorance, their prejudices, and their wants ; all which are 
unavoidable in the attempt radically to change or to originate a 
totally new system of weights and measures. 

In the law given on Sinai, not of a human legislator, but of 
God, there are two precepts respecting weights and measures ; 
first, Ye shall do no unrighteousness in judgment in mete-yard, 
measure of lengths, in weights, or in measure of capacity. 
Just balance, just weight, a just ephah, and a just hin shall 
ye have : thou shalt not have divers of weights, great and 
small, but thou shalt have perfect and just weights and meas- 
ures with all. 

Among the nations of modern Europe there are two, who by 
their genius, their learning, their industry, and their ardent and 
successful cultivation of the arts and sciences, are distinguished 
not less than the Hebrews, from whom they have derived many 
of their civil and political institutions. From these two nations 
the inhabitants of the United States are chiefly descendants ; 
and from one of them we have all our existing weights and 
measures. Both of them, for a series of ages, have been 
engaged in the pursuit of a uniform system of weights and 
measures. The legislators and all exertions by n\an have been 



THE MEASURE OF THE CIRCLE. 109 

extended to effect this, but without success, for the want of a 
perfect measure of the circle. 

It appears that a reformation is in agitation in Spain, to cor- 
rect weights and measures. It is now under the consideration 
of the courts in that kingdom ; and, as weights and measures 
are the necessary and universal instruments of commerce, no 
change can be effected in the system of any one nation without 
affecting all those with whom there are any relations of trade 
and commerce. The results of this inquiry in Spain are not 
known. France and England are the only two nations of mod- 
ern Europe that have taken much interest in the organization 
of a new system, or attempted a reform for the avowed pur- 
poses of uniformity. The proceedings with these two nations 
have been numerous, elaborate, persevering, and in France 
especially comprehensive, profound, and systematic. In both 
the phenomenon is still exhibited, that, after many centuries of 
study, of invention of laws and of penalties, almost every vil- 
lage in the country is in the habitual use of different weights 
and measures, which diversity is infinitely multiplied by the 
fact that in each country, although the quantities of the weights 
and measures are thus different, their denominations are few in 
number, and the same names, as foot, pound, ounce, bushel, 
pint, &c, are applied in different places, and often in the same 
place, to quantities altogether diverse. 

In England, from the earliest records of parliamentary his- 
tory, the statute books are filled with ineffectual attempts of the 
legislator to establish uniformity of the origin of their weights 
and measures. The very words of a law of William the Con- 
queror are cited by modern writers on the English weights and 
measures. Their import is, "We ordain and command that 
the weights and measures be as established and commanded by 
our worthy predecessors." One of the principal objects of 
the great charter was the establishment of uniformity of weights 
and measures, not of identity, but of proportion, in order to 
shut the door of dishonesty against the evasions of good rules 
and law. 

10 



110 THE MEASURE OF THE CIRCLE. 

The object of whole statutes was to guard against the fraud 
and oppression that were used with the people, for the want of 
a perfect measure. It has been remarked by all historians that 
the taking of kernels of wheat as a standard of measure is 
inconsistent with sound philosophy, as the variations of climate, 
■the seasons, and the grains in the same field, are different. 

From the year 1757 to 1764, in the years 1789 and 1790, 
and from the year 1814 to the present time, the British Parlia- 
ment has at three successive periods instituted inquiries into the 
condition of their own weights and measures, with a view to 
the reformation of the system, and to the introduction and 
establishment of greater uniformity. These inquiries have 
been pursued with ardor and perseverance, assisted by the skill 
of their eminent artists, by the learning of their distinguished 
philosophers, and by the contemporaneous admirable exertions, 
in the same cause of uniformity, of their neighboring and rival 
nations. 

Nor have the people or the Congress of the United States 
been regardless of the subject, since our separation from the 
British empire. In their first confederation, the associated 
states, and in their present national constitution, the people, 
that is, on the only two occasions upon which the collective 
voice of this whole Union, in its constituent character, has 
spoken, the power of fixing the standard of weights and meas- 
ures throughout the United States has been committed to Con- 
gress. Elaborate reports, one from a committee of the Senate, 
in 1793, another from the House of Representatives, at a recent 
period, have since contributed to shed further light upon the 
subject ; and the call of both houses, to which the report is the 
tardy but yet too early answer, has manifested a solicitude for 
the improvement of the existing system equally earnest and 
persevering with that of the British Parliament, though not 
marked with the bold and magnificent characters of the concur- 
rent labors of France, 

To despair of human improvements is not more congenial 
to the judgment of sound philosophy than to the temper of 



THE MEASURE OF THE CIECLE. Ill 

brotherly kindness. Uniformity of weights and measures is 
and lias been for ages the common/ earnest, and anxious pursuit 
of France, of Great Britain, and, since their independent exist- 
ence, of the United States. To the attainment of one object, 
common to them all, they have been proceeding by different 
means and different ultimate ends. A single and universal sys- 
tem can be finally established only by a general convention, 
where all the world shall be parties, and by one continual exer- 
tion to effect a system of uniformity of weights and measures 
that shall be meet and just to all. 

New Hampshire and Vermont. 

These two states have modelled their weights and measures 
from Massachusetts. The first act of New Hampshire to that 
effect was of the loth of May, IT 18 ; and the last, December 
loth, 1797. The statutes of Vermont have established the 
troy weight by law ; but each and every state differs in its 
measures, no two being alike. 

Rhode Island 

Has no statute on the subject. Her standards are taken ibom 
Massachusetts, and yet differ in all weights and measures. 

Connecticut. 

In the laws and standards of this state there are peculiarities 
deserving of remark. Their statute of 1800 provides that the 
half bushel shall contain 1099 cubic inches. This varies from 
some other states 29 cubic inches in half a bushel. The wine 
gallon shall be 282 cubic inches, beer measure shall be 524 
inches. 

New York. 

Xew York was originally the seat of a colony from the Neth- 
erlands, the settlers of which doubtless brought with them the 
weights and measures of their country. Towards the close of 
the seventeenth century it fell into the hands of the English ; 



112 THE MEASURE OF THE CIRCLE. 

and on the 19th of June, 1703, an act of the colonial legisla- 
ture established all the English weights and measures, accord- 
ing to the standards in the exchequer. This act was drawn 
with great care, and evidently with the purpose of embracing 
all the provisions of the then existing English statute regulat- 
ing weights, and measures, and casks, particularly those of 
1266, 1304, 1439, and 1496, without being aware of the utter 
incompatibility of those statutes with one another. 

Instead, however, of adopting in terms the London assize of 
casks, from the tun of 252 gallons downward, this act prescribes 
in inches the length and head diameter of the various casks, 
and, by a very remarkable peculiarity, changes the names of all 
dry casks. 

It directs that the hogshead shall be 40 inches long, 33 
inches in the bilge, and '27 in the head; the tierce, 36, 27, 23 ; 
the barrel, 30, 25, 22 ; half barrel, 25, 20, 16 ; quarter barrel, 
20, 16, 13. But it adds that tight barrels shall contain 3l£ 
gallons, wine measure, or with half a gallon more or less, and 
all other casks in proportion. This last provision adopted the 
whole London assize for tight casks. But the dimensions pre- 
scribed for the hogshead give the cask about 126 gallons, which 
in the London assize made the butt or pipe ; and thus the New 
York tierce was of 80 gallons, which constituted the real con- 
tents of the London puncheon ; the New York barrel was of 
60 gallons, answering to the London hogshead ; and the New 
York half barrel of 30 gallons to the London barrel. On the 
10th of April, 1784, the Legislature of New York passed an 
act to ascertain weights and measures within the state. It 
declares the standard of weights and measures which were in 
the custody of William Hardinbrook, public sealer and marker 
in the city and county of New York at the time of independ- 
ence, which were according to the standard of the exchequer, 
to he standards throughout the state. William Hardinbrook 
was directed to deliver them to the clerk of the city and county 
of New York, and to make oath that they were the same that 
he had received of the court of the exchequer. By an act of 



THE MEASURE OF THE CIRCLE. 113 

the 7th. of March, 1788, the standard weight of wheat was 60 
pounds net to the bushel. On the 24th of March, 1809, an 
act was passed relative to a standard of long measure, and in 
1813 another act. 

New Jersey. 

In New Jersey, which was originally a part of the Dutch 
settlement, the English standard of weights and measures was 
established at a later period than in New York. An act of the 
colonial legislature of the 13th of August, 1725, recites in the 
preamble that nothing is more agreeable to common justice and 
equity than that throughout the province there should be one 
just weight and balance, one true and perfect standard of meas- 
ures, for want whereof experience has shown that many frauds 
and deceits had happened ; for remedy of which it established, 
in the first section, an assize cask for the packing of beef and 
pork, since altered. The second declares that there shall be 
one just and equal balance, one certain standard for weights, 
that is to say, for avoirdupois and troy weight, one standard 
measure for the half bushel, peck, and half peck, and one for 
liquid measure ; all which shall be according to the standard of 
Great Britain in the exchequer. 

This appears to be the standard of New Jersey, but there is 
a general variation in all their weights and measures, which 
differ in size and weight from many others. They have had 
many changes in relation to the assize of barrels. 

Pennsylvania. 

In the year 1700, two laws relating to weights and measures 
were enacted by the colonial Legislature, directing that a brass 
standard from the exchequer of England should be the stan- 
dard. The second act not only adopted the London assize of 
cask, but required that all tight casks for beer, ale, cider, pork, 
beef, oil, and all such commodities, should be made of good, 
sound, well-seasoned white oak timber, and contain, — the 
puncheon, 84 gallons, the hogshead, 63 gallons, the tierce, 42, 
10* 



114 THE MEASURE OF THE CIRCLE. 

the barrel, 3l£, the half barrel 16 gallons ; wine measure to be 
according to the practice of the neighboring colonies. 

As in all other states, there is a continual variation in their 
weights and measures, for the want of a perfect measure, the 
state authorizing the innkeepers to sell beer by wine measure. 

Delaware. 
In 1705 an act for regulating weights and measures was 
passed, directing that each county should obtain standard brass 
weights and measures, according to the queen's standard for 
the exchequer. The half bushel was to be taken from Phila- 
delphia. It authorizes the standard of England. The same 
variation exists in the measures of this state from all others. 

Maryland. 

This state has had all kinds of legislative acts upon weights 
and measures, varying, in all kinds of measurement, from most 
other states. The peculiar constructions upon weights and 
measures in this state are numerous, and wonderful in the 
extreme. 

Virginia. 

Among the earliest records of the General Assembly of the 
colony of Virginia is an order of the 5th of March, 1623—4, 
that there should be no weights and measures used but such as 
should be sealed by officers appointed for that purpose. In 
this state they have had all kinds of measure, varying in every 
form and manner that ingenuity could devise. 

North Carolina. 
The only law of this state relating to weights and measures, 
a knowledge of which has been obtained, was enacted prior to 
the American revolution, during the administration of Governor 
Gabriel Johnson, and is yet in force. It prohibits the use in 
trade, by all the inhabitants or traders within the province, of 
any weights and measures other than are made and used accord- 



THE MEASURE OF THE CIRCLE., 115 

ing to the standard in the English exchequer. It charges the 
justices of the county courts to provide, at the expense of each 
county, the standards. Some of their measures are singular ; 
oil, beer, wine, bushels, pecks, are the same ; and all measure- 
ment is after this manner, varying with themselves, and every 
body, and every thing. 

South Carolina. 

By an act of the 12th of April, 1768, the public treasury 
was required to procure, of brass, or some other metal, one 
weight of 50 pounds, one of 25, one of 14, two of 6 pounds, 
two of 4 pounds, two of 2 pounds, and one of 1 pound, avoir- 
dupois weight, according to the standard of London, and one 
bushel, one half bushel, one peck, and one half peck, accord- 
ing to the standard of London. These were to be stamped 
or marked in figures denominating their weights and measures, 
and were declared to be the standard by which all others in 
the province were to be regulated. 

One continual variation exists in this state, as in all others ; 
dry and wet measures are used as the same. 

Georgia. 

An act of the Legislature of the 10th of December, 1803, 
declares the standard of weights and measures established by 
the corporation of the cities of Savannah and Augusta to be 
the fixed standard of weights and measures within the state ; 
and that all persons buying or selling shall use that standard 
until the Congress of the United States shall make provision on 
that subject. It directs the justices of the inferior courts in 
the counties to obtain standards conformable to those of the 
corporation of one of those cities. 

An ordinance of the city council of Augusta directs that all 
weights for weighing any articles of produce or merchandise 
shall be of the avoirdupois standard weight ; and all measures 
of liquor, whether of wine or ardent spirits, of the wine meas- 
ure standard ; and all measures for grain, salt, or other articles 



116 THE MEASURE OF THE CIRCLE. 

usually sold by the bushel, of the dry or Winchester measure 
standard. And it prohibits the use of any other than brass or 
iron weights, thus regulated, or weights of any other descrip- 
tion than those of 50, 25, 14, 7, 4, 2, 1, £, and £ pounds, 2 
ounces, 1 ounce, and downward. 

Kentucky. 

An act of the Legislature, of the 11th of December, 1T98, 
stating in its preamble that Congress is empowered by the fed- 
eral constitution to fix the standard of weights and measures, 
and that they had not passed any law for that purpose, recog- 
nizes as thereby remaining in force within that commonwealth 
the act of the General Assembly of Virginia of the year 1734. 
It therefore authorizes and directs the governor to procure one 
set of the weights and measures specified by the Virginia act 
of 1784, with measures of the length of one foot and one yard, 
and declares that the bushel, of dry measure, shall contain 
£150| solid inches, and the gallon, of wine measure, 231 
inches. 

Here is a difference in the bushel of 106 inches from some 
states, and in the wine measure of 10 inches in one gallon from 
some other states under this government. 

Tennessee. 

From a communication received from the governor of the 
State of Tennessee, it appears that there is in that State no 
standard of weights and measures fixed by the Legislature. 

Ohio. 

The only act of the Legislature of the State of Ohio on this 
subject is of the 22d of January, 1811. It directs the county 
commissioners of each county in the State to cause to be made 
one half bushel measure, to contain 1075t 2 <j inches, solid, which 
is to be kept in the county seat, and be called the standard. 



THE MEASURE OF THE CIRCLE. 117 

Louisiana. 

Before the accession of Louisiana to the Union of these 
States, the weights and measures used in the provinces were 
those of France, of the old standard of Paris. An account of 
these, and of the present state of the weights and measures in 
the State of Louisiana, is submitted in the appendix to this 
report. This State uses for a barrel 330 cubic inches less than 
some other States, all under one government. 

Indiana. 

An act of the territorial Legislature of the 17th of Septem- 
ber, 1807, authorized the courts of common pleas of the 
respective counties in the territory, whenever they might think 
it necessary, to procure a set of measures and weights for the 
use of the country, — one measure of 1 foot, or 12 inches, 
English measure, so called, one measure of 3 feet, or 36 inches, 
English measure, and half a bushel, of dry measure, to contain 
10753- solid inches, and one gallon measure, to contain 231 
inches. 

This State differs in the dry measure from some of the States 
53 inches in the bushel; other measures in the same propor- 
tion, or more. 

Mississippi. 

An act of the territorial Legislature, of the 4th of February, 
1807, directed the treasurer to procure a set of the large avoir- 
dupois weights, according to the standard of the United States, 
if one were established, but if there were none such, according 
to the standard of London, with proper scales for weights, 
together with measures of the foot and yard, dry measure of 
capacity, and liquid wine measure. 

This standard is to remain until the United States fixes a 
standard. A barrel of flour contains 196 pounds; pork and 
beef, 200 pounds to the barrel. 



118 THE MEASURE OF THE CIRCLE. 



Illinois. 



The territorial act of the 17th of September, 1807, was 
passed while the State of Illinois formed a part of the Indiana 
territory ; but by the act of the Legislature of this State, regu- 
lating weights and measures, of the 22d of March, 1819, the 
county commissioners of each county in the State were required 
to procure, at the expense of the county, one foot, one yard, 
English measure, a gallon liquid or wine measure, to contain 
23 1 cubic inches, a corresponding quart, pint, and gill measure, 
of some proper metal, a half bushel, dry measure, to contain 
eighteen quarts, one pint, and one gill, wine measure, or 1075^- 
cubic inches, and a gallon, dry measure, to contain one fourth 
part of half a bushel ; these measures to be copper or brass. 
Also, weights, one pound, J pound, £, and T V, made of brass, 
the integer of which to be denominated one pound, avoirdupois, 
and to equal in weight 7020 grains, troy or gold weight. 

The most remarkable peculiarity of this act is, its departure 
from the English standard weights by fixing the avoirdupois 
pound at 7020 instead of 7000 grains troy. 

Alabama. 

This State having formed a part of the Mississippi territory 
previous to the admission of the State of Mississippi into the 
Union, in 1817, the acts of that territory of the 4th of Febru- 
ary, 1807, and the 23d of December, 1815, embraced this sec- 
tion of the territory. No act of the State Legislature of Ala- 
bama on this subject is known to have been passed. 

Missouri. 
The territorial Legislature, by an act of the 28th of July, 
1813, directed the several courts of common pleas within the 
territory to provide, at the expense of the respective counties, 
one foot, one yard, English measure, one half bushel, to con- 
tain 1075| solid inches, for dry measure, one gallon, to contain 
231 inches, and smaller liquid measures in proportion, to be of 



THE MEASURE OF THE CIRCLE. 119 

any wood or any metal the court think proper ; also, one set of 
avoirdupois weights. The use or keeping, to buy or sell, of 
weights or measures not corresponding with these standards, 
after due notice, was prohibited, under penalties, by the same 
act, but with a provision that all contracts or obligations made 
previous to the taking effect of the act should be settled, paid, 
and executed, agreeably to the weights and measures. 

District of Columbia. 

By the act of Congress of the 27th of February, 1801, con- 
cerning the District of Columbia, the laws of the State of Vir- 
ginia, as they then existed, were continued in force in the part 
of the district which had been ceded by that State, and the laws 
of Maryland in the part of the district ceded by Maryland. 

The act to incorporate the inhabitants of the city of Wash- 
ington, of the 3d of May, 1802, authorizes the corporation to 
provide for the safe keeping of the standard of weights and 
measures fixed by Congress, and for the regulation of all meas- 
ures used in the city. The supplementary act of the 24th of 
February, 1804, gives the city council power to establish and 
regulate the inspection of flour, tobacco, and salted provisions, 
and the gauging of casks and liquor. 

There have been in this District numerous changes and alter- 
ations. 

As preliminary remarks in reference to that part of the reso- 
lutions of both houses which requires the opinion of the Secre- 
tary of State with regard to the measure which it may be proper 
for Congress to adopt in relation to weights and measures, I 
will state what might be done in Congress : in one line of the 
constitution to adopt the French standard. To fix the standard 
appears to be an operation entirely distinct from changing the 
denominations and proportions already existing, and established 
by the laws, or immemorial usage. In Europe, every historical 
research presents a fruitless struggle to effect a uniform system 
of weights and measures ; less exertion, perhaps, exists in the 



120 THE MEASURE OF THE CIRCLE. 

United States for the weights and measures than elsewhere. 
The United States adopted their measures and weights from 
England, or from usage ; and no general exertions have taken 
place, like those of the old country. The wine gallon of 231 
inches, and the beer gallon of 282 inches, have been known by 
usage, and the Winchester bushel of 2150.42 inches formed 
the general standard. The various multipliers of the yard, ell, 
perch, pole, furlong, acre, and mile were recognized by law. 

The experience of the French nation under the new system 
has already proved that neither the immutable standard from the 
circumference of the globe, nor the isochronous vibration of 
the pendulum, nor the gravity of distilled water at its maximum 
of density, nor the decimation of weights, measures, moneys, 
and coins, nor the unity of weights and measures of capacity, 
nor yet all these together, are the only ingredients of practical 
uniformity for a system of weights and measures. It has 
proved that gravity and extension will not walk together with 
the same staff; that neither the square, nor the cube, nor the 
circle, nor the sphere, nor the gravitation of the earth, nor the 
harmonies of the heavens, will gratify the pleasure, or, to 
indulge the indolence of man, be restricted to computation by 
decimal numbers alone. 

A perfect measure of the circle may be considered as enter- 
ing into the economical arrangements and daily concerns of 
every family. It is necessary to every occupation of human 
industry, to the distribution and security of every species of 
property, to every transaction of trade and commerce, to the 
labors of the husbandman, to the ingenuity of the artificer, to 
the studies of the philosopher, to the researches of the antiqua- 
rian, to the navigation of the mariner, and the marches of the 
soldier ; to all the exchanges of peace, and all the operations 
of war. It is among the first researches of literature; the 
establishment of its truth is among the first elements of educa- 
tion. This knowledge is riveted in the memory by the habitual 
application of it to the employments of men, through life. 
Every individual, or, at least, every family, has the weights 



THE MEASURE OF THE CIRCLE. 121 

and measures used in the vicinity, and recognized by the cus- 
tom of the place. To change all this at once is to affect the 
well being of every man, woman, and child in the community. 
It enters every house, it cripples every hand. 

"Weights and measures, and the final establishment of a sys- 
tem for them, with a view to the utmost practical extent of uni- 
formity, are at this moment under the deliberate consideration 
of four populous and commercial nations, — Great Britain, 
France, Spain, and the United States. The interest is common 
to them all ; the object of uniformity is the same to all. Could 
they agree upon one result, the advantages of that agreement 
would be great to each of them, and still greater in all their 
intercourse with one another. It is therefore respectfully pro- 
posed, as the foundation of proceedings necessary for securing 
ultimately to the United States a system of weights and meas- 
ures which shall be common to all civilized nations, that the 
President of the United States be requested to communicate, 
through the ministers of the United States in France, Spain, 
and Great Britain, with the governments of those nations, upon 
the subject of weights and measures, with reference to the prin- 
ciple of uniformity, as applicable to them. 

The Pendulum. — It is proposed to discard all considerations 
of the pendulum, as the theory of its vibration, however inter- 
esting in itself, is believed to be, since the definitive determina- 
tion of the metre, useless with reference to any system of 
weights and measures. 

The most prominent numbers made use of in the Bible are 
3, 6, 9, and 12. The number 6 prefigures time, and measures 
the circle, with the hexagon, the hexagon being equal in all its 
parts. These numbers — 3, 6, 9, and 12 — are used in the 
performance of all principal important things where numbers 
are used — 12 apostles, 12 signs in the zodiac, and 3 persons 
in the Godhead, Father, Son, and Holy Ghost. 
11 



122 



THE MEASURE OF THE CIRCLE. 



MEASURE BY CUSTOM HOUSES IN THE UNITED STATES. 

The following are the contents of the different dry measures, 
by admeasurement, and by the weights of water they contain, 
as obtained at the different custom houses by the investigation 
directed by the late President John Q. Adams, reduced to the 
bushel derived from each : — 



Names of the Custom Houses 






Cubic Inches. 


Avoird 
Lbs. 


jpois Weight 
oz. dwt. 


Bath, Me., ...... 1925.00 . 


. . 74 


2 


Belfast, . . 






. 2063.76 . 


. . 76 





Frenchman's Bay, . 






. 2216.70 . 


. . 84 


7 8 


Kennebunk, . 






. . 2203.32 . 


. . 78 





Machias, .... 









. . 75 


4 


Lubec, . . . 






. 2158.33 . 


. . 78 


14 


Portland, . . 






. 






Falmouth, . . 






. 






Saco, .... 






. . 2215.80 . 


. . 80 





Wiscasset, 






. 






Portsmouth, N. H., 






. . 2153.74 . 


. . 77 


12 


Boston, . 






. 2211.06 . 


. . 78 


4 


Newburyport, . 






. . 2150.52 






Gloucester, . 






. 2150.40 . 


. . 78 


8 


Dighton, . . . 






. 2062.78 . 


. . 75 


13 


New Bedford, . 






. 2155.12 . 


. . 77 


13 8 


Barnstable, . 






. . 2153.82 . 


. . 77 





Edgartown, . 






, . 






Nantucket, . . 






, 






Providence, K. I., 






. . 2194.50 . 


. . 78 


8 


Bristol, . 






. 2155.13 . 


. . 78 





Newport, . 






. . 2160.18 . 


. . 77 


14 


New London, Ct., 






. . 2222.06 . 


. . 78 


10 


Fairfield, . . . 






. . 2249.86 . 


. . 79 





New York, mean, 






. . 2152.56 . 


. . 78 


13 4 


Rochester, N. Y., 






. 2202.58 






Philadelphia, 






. . 2186.20 . 


. . 78 


12 



THE MEASURE OF THE CIRCLE. 



123 



Names of the Custom Houses. 


Cubic Inches. 


Avoirdupois Weight. 
Lbs. oz. dwt. 


Wilmington, Del., . 


. 2192.20 . . 


. 77 


4 12 


Baltimore, Md., . . . 


. 2150.42 . . 


. 77 


8 


Oxford, Md., . . . 


. 2060.94 . , 


. 80 





Washington, D. C, 


. 2117.20 . 


. . 76 


7 10 


Georgetown, D. C, 


. . 2152.60 . 


. . 77 


14 2 


Alexandria, .... 


. 2118.80 . 


. 77 


11 


Cherrystone, .... 


. . 2225.48 . 


. . 83 


4 


Norfolk, Va., . . . 


. . 2127.24 . 


. . 78 





Petersburg, . ... 


. . 2147.08 . 


. . 78 





Richmond, .... 


. 2112.60 . 


. . 78 


8 


Camden, N. C, . . . 


. . 2152.20' . 


. . 79 


8 


Edenton, 


. . 2160.78 . 


. . 77 


6 


Newburn, .... 


. . 2115.60 . 


. . 87 


8 


Ocracoke, 


. . 2153.10 . 


. . 76 





Plymouth, 


. 2358.58 . 


. 77 





Washington, N. C, . . 


. 2128.02 . . 


. 72 


12 


Charleston, S. C, . 


. . 2172.03 . 


. . 77 


12 12 


Savannah, Ga., . . . 


. 2013.32 . 


. . 76 





St. Mary's, Ga., . . . 


. 2019.34 . 


. . 78 


4 


New Orleans, . . . 


. 2162.02 . 


. . 77 


11 7 



In the United States, their coins, both gold and silver, are 
legal tender for payment, to any amount; but in England, 
silver coin is a legal tender for payment only to an amount not 
exceeding 40 shillings; and by the restrictions of each pay- 
ment by the bank, the only actual currency, the only material, 
in which an American merchant having a debt due to him in 
England can obtain payment is Bank of England paper; so 
that at this time the material of exchange between the United 
States and England is, on the side of the United States, gold or 
silver, and on the side of Great Britain, bank paper. Suppose 
an American merchant has a debt due him in England, which 
is remitted to him in gold bullion, or coin of the English stan- 
dard, say £10,000. He receives of pure gold 196 pounds, 2 
ounces, 3 pennyweights, 22 grains, for which, when coined at 



124 THE MEASURE OF THE CIRCLE. 

the mint of the United States, he receives 45,657 dollars, 20 
cents. The pound sterling therefore yields him 4 dollars, 
56.572 cents, which is the value of a pound sterling if the par 
of exchange be estimated in gold, according to the standard of 
purity common to both countries. If the payment should be 
made in silver bullion, at 66 shillings the pound, troy weight, 
according to the present English standard of silver coinage, he 
would receive only 43,489 dollars and 43 cents ; and the pound 
sterling would only net him 4 dollars, 34.8943 cents. The 
pound sterling, estimated in gold, is worth 4 dollars, 56.5720 
cents ; in silver it is worth 4 dollars, 34.8943 cents ; making a 
difference of 21.6777 cents, half of which, 10.8388, added to 
$4.348943, and deducted from $4.565720, makes what is called 
the medium par exchange, $4.457331. 

One pound, troy weight, of uncoined gold, or foreign gold 
coin, eleven parts fine and one part alloy, is $209.77 ; one 
pound of silver, eleven parts fine and one part alloy, is $13.77. 
In April, 1816, an act was passed, regulating the currency, 
within the United States, of the gold coins of Great Britain, 
France, Portugal, and Spain, the crowns of France, and 5 -franc 
pieces. By this act, gold coin of Great Britain and Portugal 
weighing 27 grains equal 100 cents, or 88f cents per dwt. ; 
France, 27£ grains equal 87£ cents ; Spain, 28J grains equal 
84 cents; crowns of France weighing 449 grains equal 110 
cents, or $1.17 per ounce; 5-franc pieces weighing 386 grains, 
93.3, equal $1.16. 

One pound, troy weight, of standard gold in England con- 
tains 5280 grains of pure gold. It is coined into £46 14 s. 
11.214 d. Then 11.214 : 5280 : : 240 : 113.0014 grains of 
pure gold in a pound sterling. 

In the United States, 24.75 grains of pure gold are coined 
into a dollar, or 247.5 grains to one eagle. Thus 24.75 : 1 : : 
113.0014 : $4.56572 to 1 pound. Thus the pound sterling in 
gold is worth $4.56572; and as 5280 : 11.214 : : 24.75 : 
52.5656 dollars in English gold, 4 s., 4.5656 pounds sterling 
in gold, $4.56572, 



THE MEASURE OF THE CIRCLE. 125 

One pound, troy weight, of standard silver, in England, con- 
tains 5328 grains of pure silver, and is coined into 66 shillings, 
or 792 pence. The dollar of the United States contains 371.25 
grains of pure silver. Then 5328 : 792 : : 371.25 : 55.1858 
dollars in English silver, 4 s. 7.1858 d. ; 792 : 5328 : : 240 : 
1614.545 grains of pure silver in a pound; 371.25 : 1614.545 
: : £1 4.348943s. sterling in $4.348943, silver; medius par 
dollar, 4 s. 5.8757 pence. £1 sterling in gold, $4.565720 
— 10.8388 = $4.457331, med. par; £1 sterling in silver, 
$4.348943 +. 

"We are told that 1800 circumference will not work in pro- 
portion as 6 to 19, as other numbers. 



If 19 : 6 : : 1800 
6 


If 18: 


1 : : 568* 
1 


19 ) 10800 ( 568* 
95 


18) 568* (3 Hi 
54 


130 
114 


" 


28 

18* 


160 
152 


10 

Multiply by 19 


8 
I add to the 568* the 3l|j. 

600 

This is the diameter of 1900. 




90 
10 

198M 

18 

18 
18 



If 600 : 1900 : : 568*, answer 1800. 

I now find the area of 568*. 
11* 



126 



THE MEASURE OF THE CIRCLE. 



568.421 
568.421 

568421 
1136842 
2273684 
4547368 
3410426 
2842105 

323101.433241 Take J of this. 

7 



9)2261710.032631 



I divide this bv 28. 28 ) 251301.113631 ( 8975.09772 

224 



251301.113631 
8975.039772 

3)242326.073859 

V 80775.357953(284.210 



252 

210 
196 

141 

140 



284.210, radius, 
o 



568.420, diameter. 
I now take the diameter, 
and find its circumfer- 
ence. 284x2=568, 
diameter. 568 

9.5 



This is the 
radius. 



48)407 
384 



564)2375 
2256 



111 

84 

273 
252 

216 
196 



2840 
5112 

3)53960 

1798.6 

9, 



5682)11935 
11364 



56841)57179 
56841 



33853, remainder. 



203 1800 

196 I add the 2 

for the T 8 g-, 

71, circum.] and call 
56 the remainders 

— equal to the 2 

15, rem. remaining. 



THE MEASURE OF THE CIRCLE. 127 

In the first place I show that 6 to 19 brings the diameter of 
a circle of 1800 to be 568 T % ; then I show that T V is 31y£; 
now add this to the 568 T 8 ^, which gives 600. This 600 is the 
diameter of 1900 ; then I show that if 600 gives 1900, 568 T V 
will give 1800 ; then I find the area of the 568 T % ; then I find 
the circle that bounds the area, which is 1800. 



568.421 T V 
9.5 



2842105 
51157895 

3 ) 54000000 

1800.0000 



The rule to find the circle is, to divide by 28, subtract the 
quotient, divide by 3, get the square root, and that will be the 
radius of the circle that bounds the figures. 

On the following page I have given the working of 1500 cir- 
cumference. I have worked it as far as it is possible, because it 
begins to repeat, as you may see. It begins with 90, and comes 
to 90 again. It is what is called a surd ; and it is said its root 
can never be perfected. I say it cannot be perfected by deci- 
mals, but by a vulgar fraction it can, as 4 pence are just as 
perfectly J of a shilling, as T 5 g- are equal to £, except that one is 
a vulgar fraction, and the other a decimal. 

First, what is the diameter? — 



128 THE MEASURE OF THE CIRCLE. 



If 19 


:6: 
19] 


: 1500 
6 




9000 ( 473.68421052631* 

76 






~140 
133 






"To 

57 






"lio 

114 






"l60 
152 


I find the diameter more in 
whole numbers : — 


80 

76 


If 19 : 6 : : 1500 

6 




~40 

38 


19) 9000 ( 473 U 
76 


~20~ 
19 


140~ 
133 




100 
95 


70 
57 




~~50 
38 


"TF 




120 
114 




60 
57 






~30 
19 






110* 
95 






150 
133 






170 
152 




180 
171 



90 



THE MEASURE OF THE CIRCLE. 129 

473 
13 

1419 
473 

19 } 6149 ( 323 X 2 = 647, plus. 

57 * 

473 44 
473 38 

1419 69 

3311 57 

1892 — 

447, added. 12 



8, for the corner. 



224376 — 

7 20 



9 ) 1570632 ( 174914f+ This is the area of 1500 
9 circumference. 



67 
63 



40 
36 



46 
45 



13 
9 

42 
36 



6 This remainder is equal to J. 

To find the measure of a globe or sphere, or, rather, if a 
certain number of miles, feet, inches, or whatever you may 



130 THE MEASURE OF THE CIRCLE. 

please to call it, be made to compose a globe, what the diameter 
of that globe will be : Suppose 8064 to be the given number ; 
I now divide 8064 by 7, and multiply that quotient by 5, the 
product of which I add to the given number, thus : — 

8064 
5760 



13824 



I now extract the cube root of this number, and the quotient 
will be the diameter of the globe. 

7 ) 8064 



1152 
5 



5760 Add this to the given number, 8064. 



V* 13824 I find the cube root of this. 

13824 ( 24 is the cube or diameter of globe. 
8 

5824, first resolved. 

6, three times the root. 
12 three times the square of the root. 

126, the first divisor. 

64, the cube of 4. 
96 square of 3 multiplied by 3 times 2. 
48 three times the square of 2 by 4. 



5824, subtrahend. 



0000 



Proof: tV of the cube is the globe, and the cube is 13824 ; 
therefore, divide the cube by 12, and 7 times the quotient is 
the globe : — 



THE MEASURE OF THE CIRCLE. 131 

12 ) 13824 



1152 

7 



8064, answer. 

Having examined Messrs. Hodson and Pike's works, I find 
that in extracting the cube root, having found the divisor, they 
place one less than it will go, which is right in the sum laid 
down; but I think they ought to have given reason for so 
doing, otherwise it is a great puzzle. Now, in the sum on the 
preceding page it gives just the number of times ; but in that 
laid down by Pike it goes one less, thus : 

V 3 16194277 ( 253 

8 

8194, subtrahend. 

126, divisor. 

116 
216 

72 



9576, subtrahend. 



125 

Thus. 150 
60 

7625, right subtrahend. 

A GENERAL VIEW OF THE WORK. 

I show by the strips of paper, which are 9 inches long, that 
the vacant corners are the biquadrate of the circle. The biquad- 
rate is the 4th power ; it is the T V of the hexagon, or the tV 
of 3 times the diameter added, which completes the measure 
of the circle. 



132 THE MEASURE OF THE CIRCLE. 

The area of trie dodecagon can be found by the six oblong 
squares, as laid down in my work ; but it does not find the cir- 
cular part outside of the dodecagon, which is just the square 
of the biquadrate, and that added will complete the area of any 
circle, which is the 27th part of the dodecagon, added to the 
dodecagon makes a 28th part of the whole. This biquadrate 
is the 4th power of the number 6 ; but a biquadrate is the 4th 
power of any number. 

In a circle 6 inches in diameter the biquadrate gains 1 inch 
upon a line ; therefore the square of that inch will make just 
one square inch in the area ; whereas a circle 12 inches in diam- 
eter gains 2 inches, linear measure, making just 4 square inches 
in the area, as 2 X 2 = 4. 

The square of what the biquadrate gains in circumference is 
what the circle gains in area ; so that the 4 inches is just what 
is contained without the dodecagon, which is the circular part 
of the circle, and is what makes up the measure of the circle. 
The biquadrate descending is 1296 parts of one inch; the 
biquadrate in a 12 inch circle is 2 inches on a line, and the 
square of that is the area, which is 4. 

I show that I find the area of a circle by dividing the circle 
into 6 angles, and forming an oblong square from one of them, 
equal to the product of the radius by half the radius, which 
finds the square inches, and by adding the biquadrate to the 
square inches, makes up the measure of the circle. 

I am asked what evidence I have to prove that the propor- 
tion the diameter of a circle has to its circumference is as 6 to 
19 ? I answer, there is no other way to prove that an apple is 
sour, and why it is so, than by common consent. For exam- 
ple : I will suppose that the diameter of a circle to its circum- 
ference is as 6 to 19. I then suppose a circle 12 inches in 
diameter to be filled with rings £ °f an mcn wide, and 36 in 
number. I suppose the outside ring to be on its outside 38 
inches. I find that all these rings measure 114 inches, if I 
measure them in straight strips, and square at the ends ; the 
average length of the rings on a straight line is 19 inches. 



THE MEASURE <3E THE CIRCLE. 133 

Now, to form a circle of these rings, I take f of an inch 
from the inside of the ring, which goes to the length to form 
the circle. I then find that these 36 rings, -g- of an inch wide, 
measure 112 inches ; and as my diameter is 12 inches, and my 
longest ring is allowed to be 38 inches on the outside, I find 
the proportion is as 6 to 19 ; as 3 times 12 are 36, and 2 are 
38 ; and 3 times 6 are 18, and 1 are 19 ; and 4 times 9 are 36, 
and 2 are 38 ; and twice 9 are 18, and 1 are 19; and 4 times 
9 are 36, and 2 are 38. My proportion of the diameter to the 
circumference is as 6 to 19, and my ratio is 9.5. 

I defy mathematics to prove any other proportion that the 
diameter of a circle has to its circumference. Why not ask, 
with the same propriety, what proportion the square has to the 
square, when it is as 12 to 12 ; how can I prove that is the 
proportion ? If it is 144 inches, why is it a square ? 

There are 18 perfect equilateral angles in any circle, great or 
small, which is 60 degrees, and there is no other triangle that 
will be equal but 60 degrees. The curve line gains T V over 
the straight, on every side of the hexagon ; therefore the pro- 
portion must be as 6 to 19. 

A circle 6 inches in diameter is 28 inches in area ; and one 
12 inches in diameter is 112 in area. Twice the diameter is 
4 times the area, and so in proportion. 

Archimedes' measure, as 7 to 22, is just £ of an inch in 33 
too short, almost T 3 ^ in the yard, and near T V in the foot. The 
biquadrate is equal to ^ of -g-V ; and the sides of the hexagon 
being divided by 36, the two sides of the biquadrate will equal 
-s 2 6, or yV, on one side of the hexagon, and the 6 sides will gain 
2 inches, as 2 added to 36 equal 38. The average measure of 
the rings is 3% inches. Every thing in a circle has equal pro- 
portions ; it is the most perfect thing in nature ; every part 
must have a natural and proportional result. The hexagon 
gives ^ of £ for curve, in every part, and ^ of £ is T V of itself; 
so the circle gains T V of every side of the hexagon ; therefore 
it gains T V of the whole ; so if the 6 sides of the hexagon be 
12 



134 THE MEASURE OF THE CIRCLE. 

36 inches, the circle is 38, as 7 to 22 is short of correct meas- 
ure £ of an inch. 

The use of the measure of the circle is, to. find the circum- 
ference of any circle, great or small, in order to correct all 
weights and measures, which are wrong, and have been since 
the world began. No man knows what an inch is, a foot, a 
yard, a rod, a degree, a peck, a quart, a half bushel, a bushel, 
a barrel, a hogshead, a pipe, a gallon, or any other measure. 

No man can find a perfect level on the earth, for the want 
of this measure ; no man can find the contents of a steamboat, 
boiler, except by this measure ; no man can measure a piece of 
land, if in a circular form, without a perfect measure of the 
circle ; no man can tell the number of inches in a grindstone, 
without a perfect quadrature of the circle ; no man can gauge 
a cask of liquor without this measure ; no man ever did or ever 
can extract the square root of the surd number without this 
measure ; no man can give the contents of a globe or sphere 
without it. 

All circular mechanical operations labor under great disad- 
vantages, for the want of the quadrature of the circle. The 
calculations made by our astronomers of the earth's traverse 
round the sun is very imperfect, inasmuch as they make it in 
every 24 hours 1,500,000 miles ; whereas correct measure 
makes it over 1,700,000 miles. 

To show the absurdity of the expression from so called great 
men, such as G. B. Arry, professor royal of England, that the 
circle is measured near enough for any thing, I will show the 
difference between the measure of a globe or sphere, taken 
from books now in use, and the perfect measure. The present 
measure makes the number of cubic inches to be 904.7808 ; I 
make it 1008 by perfect measure. So you see the difference 
in a globe 12 inches in diameter is 103 inches, and a decimal 
Of 2192. 

The present measure makes the convex surface of a globe 12 
inches in diameter 452.3804 ; I make it 361 only. This makes 
a difference of 91 inches, and 3804 decimal. 



THE MEASURE OF THE CIRCLE. 135 

Look at this, if measure is near enough. All astronomical 
observations, which have been attended with great labor and 
anxiety, and much loss of time, can by this measure be demon- 
strated with perfection and ease. 

The difference between as 7 to 22 and as 6 to 19 is as T fe is 
to y^. This, as I have before said, makes the foot rule of 12 
inches near T V too short, and the' yardstick near ■$$. Linear 
measure is .75, square measure 1.52£, cubic measure 2.29 per 
cent, astray. The proportion the diameter has to the circum- 
ference is as 6 to 19 ; my ratio is 9.5. 

Let the cooper take three times the diameter, with one third 
of the radius, and his head will just fill. 

Take a thin strip of tin or brass, the thinner the better, cut 
a strip 38 inches long, and form a perfect circle, and the diam- 
eter will be 12 inches. 

To show the difference of area between the square and the 
circle : Cut a piece of wire 38 inches long, form a perfect cir- 
cle, and the area will be 112 inches. Cut another piece 38 
inches long, and form a square of 9^ inches, and the area con- 
tained in the square will be 90£ inches. Thus the difference 
of area between the circle and the square will be 2 If inches. 

A gentleman by the name of North, a descendant from Lord 
North, of England, contemplates building a glass globe, one 
mile in diameter, in the United States, with the principal places 
in the world painted within the globe, so that, by the construc- 
tion of seats within, the people can sit with ease, and see the 
operation of the earth, and all its principal places. How many 
square feet of glass would it require ? As St. Paul says, in his 
epistle to the Romans, chapter 11, verse 14, "I do this, if by 
any means I may provoke any of you to emulation " in measur- 
ing the circle. It will take 68,889,600 square feet, making no 
allowance for deductions. 

In a former part of my work I have given a table from John 
Q. Adams, that people may see the varieties of measures that 
are made use of, whereas they all ought to be one. These vari- 



136 THE MEASURE OF THE CIRCLE. 

eties exist only for the want of a perfect measure of the circle, 
which will be meet and just to all. 

I will here give the principal rules that are necessary for 
practical use, which may be learned by a common scholar in 
twenty-four hours. 

Suppose your circle is 12 inches in diameter ; multiply the 
diameter by 9.5, which is my ratio for finding the circumfer- 
ence ; divide the product by 3, which gives the perfect circum- 
ference in all cases. Thus : — 

12 
9.5 

108 
6 

$ 3)114' 

38, circumference. 

To find the area of the same : Take 3 times the radius by 
once the radius, which gives the square inches ; divide the , 
square inches by 3 to the 4th power or biquadrate ; add the 
4th power or biquadrate to the square, which gives the perfect 
area, in all cases. Thus : — 

6 
3 

18 108 

6 4 

3)108 112, area. 

3)36 
3)12 

4th power, or biquadrate. 



THE MEASURE OF THE CIRCLE. 137 

Or you may take £ of the square of the diameter, which will 
give you the perfect area, in all cases. This is more simple 
and more easy than the above rule. Thus : — 

12, diameter. 
12 

144 

7 



9 ) 1008 



112 



Having the circumference, to find the diameter : Suppose 
your circumference is 38 ; divide the circumference by 19, and 
multiply the quotient by 6, which gives the diameter, in all 
cases. Thus : — 

19)38(2 
38 6 

12, diameter. 

Having a promiscuous number of figures, to find the circle 
that bounds them : Divide the figures by 28, subtract the quo- 
tient from the sum ; divide the remainder by 3 ; then extract 
the square root of the quotient, which will be the radius of the 
circle that bounds the figures ; and the square root of any num- 
ber of figures wrought in this way will be the radius of any 
circle that bounds the figures. Thus : — 

28 ) 448 ( 16 448 

28 16 

"168 3)432 

. 144 ) 12, radius. 

1 

22)044 
44 







12* 



138 THE MEASURE OF THE CIRCLE. 

The radius is 12, and twice the radius is the diameter ; so the 
diameter is 24. 

I have given the principal rules both in the first and last 
pages, in order that persons may see the principal practical 
rules which will enable them to effect all operations of the 
measure. 

Equality and proportion are now demonstrated; space and 
distance, which never have been known, can now be deter- 
mined. The difference of the square and the circle can now 
be told ; the area of both is now demonstrated ; what the circle 
gains over the square is illustrated ; the difference of the two 
lines which are called arc and line is known. 

What is contained in the dodecagon is now known ; a vari- 
ety of questions which have slept in oblivion can now be math- 
ematically demonstrated, and used for the benefit, happiness, 
and pleasure of all the learned, and the lovers of science, who 
may wish to benefit mankind. 

In the course of my exertions to measure the circle I have 
been met with all kinds of objections that ingenuity and art 
could contrive. I have been told that trigonometry finds the 
measure of the circle to a hair's breadth. 

Now, this is not true. Trigonometry has never been brought 
to such perfection as to measure a circle. Its use has been to 
find distance and space, the bounds of which are already sta- 
tioned and found. Its use is also to find the measure of trian- 
gles, and their area, that is, to find three sides ; and as it has 
been used, it has never found but two, in its operation, to meas- 
ure a circle. 

I shall endeavor to show how it can be perfected by trigo- 
nometry. I take the number 6, it being the most perfect num- 
ber that I can find for my use. Now, the hexagon is composed 
of 6 equal triangles of 60 degrees each ; but for convenience 
I divide them into 36 degrees ; and as the triangle gains T V, to 
make the curve of the circle, I call it 38, two above an equal 
triangle ; and 6 times 38 equal to 38 inches for the circumfer- 
ence of 12 diameter. 



THE MEASURE OF THE CIRCLE. 139 

Every circle that is twice the diameter of another gains four 
times as much for area, just the same as the square does. Sup- 
pose a square of 3 inches ; then 3 inches multiplied by 3 give 
9, area ; then 6 inches multiplied by 6 give 36, and 4 times 9 
are 36 ; so 12 multiplied by 12 is 144, and 4 times 36 are 144. 
Now, if the circle does not come in the same proportion, it 
proves itself wrongly measured ; but if it does come in the 
same proportion, it proves itself right. 

Furthermore, if in the first place there is ever so small an 
error, the error will increase in a fourfold proportion. Thus, 
if in 3 inches diameter there be a very small error, in 6 inches 
the error will be 4 times as great, and in 12 inches 16 times, 
which proves the work, beyond all doubt. 

The area of the circle is $ of the square ; so 3 times 3 is 
equal to 9 square inches ; and £ of that is 7 inches, which is 
the area of a circle 3 inches in diameter. Now, 6 inches diam- 
eter is 4 times 7, equal to 28 ; and 12 diameter is 4 times 28, 
which is 112, the area of a circle 12 inches in diameter. And 
you may carry it as far as you please, which will prove the 
measure to perfection. 

Now, in trigonometry the circumference loses -^V by bring- 
ing the curve to a straight line, which makes the diameter in 
the centre of a triangle of 12 inches 112 instead of 114 sixths 
of an inch ; and the radius being 6 inches, every sixth in the 
centre makes 1 inch; so 112 inches is the perfect area of a 
circle 12 inches in diameter. A string that is 38 inches in 
length in a circle is -^V shorter on a straight line. 

I take trigonometry to find the area of one sixth of a circle 
12 inches in diameter, and it gives 18.66§; but by square 
measure it would be 19 inches, gaining 6 for the curve of the 
dodecagon ; but by trigonometry it makes but 4, which added 
to 6 times 18, or 108, makes 112 for area; or you may take 
the rule of three for the preceding measure of one sixth of a 
circle, thus : If 90 gives 6, what will 60 give ? Answer, 4 ; 
and 4 added to 108 is equal to 112, the area; or, as the cir- 
cumference is 38 inches, and the hexagon, or 6 times the 



140 THE MEASURE OF THE CIECLE. 

radius, measures but 36, you will see the circle gains 2 inches 
on a line ; but as the line has no area, I square it, thus : 2 X 2 
— 4 ; and this added to 108 makes 112, which proves that all 
give the same area if the diameter be 12 inches; and it also 
proves the circumference right, for as it gains just 4 inches, 
there is no other measure that would perfect the measure of the 
area ; so the one proves the other. 

Now, to find the triangle equal to the area, as the six sides 
of the hexagon measure 36 inches, it wants T \, equal to 2, 
which added to 36, make 38, the measure of the circle. But 
the circle being a curved line, it loses % of T V, equal to -£ T , of 
a straight line. So half the circumference being 114 sixths of 
an inch, subtract -V, equal 2, from 114, which leaves 112, 
equal 18 f for the base of the triangle, and the erect side is the 
radius, equal 6 inches, and the side wanting is the hypotenuse ; 
but instead of seeking it, the best way is to take the base line, 
112, with the erect line, 36, equal to 6 inches, or radius, and 
there being two triangles, reverse them, and they will make an 
oblong square, 18f inches by 6 ; multiply it together, and it 
will be the area. Thus : — 

18| 
6 

112, perfect area. 

Or you may take one half the circumference, equal 114, and 
subtract -£ Ti which leaves 112 sixths of an inch; it makes an 
oblong square, 18| by 6, the radius; multiply it, which makes 
112, area. If you do not subtract 57, you will find when you 
come to straighten the circle, to make the square, it would 
expand, and make the area too much. 

The question is asked, Can you square, the circle ? or, what 
is the square of the circle ? Answer : I take the hexagon, 12 
inches in diameter, and divide every angle into two equal parts, 
making it 12 sides, or a dodecagon, which will measure just 
108 inches, as may be proved by the oblong square, as shown 
above. 



THE MEASURE OF THE CIRCLE. 141 

So the dodecagon is no part of a circle. Now, the six sides 
of the hexagon measure just 36 inches, and it gains just 2 
inches in making it into a circle ; therefore the circumference 
is 38 inches, and the two that are gained make up the measure 
of the circle. So you see the square of 2 is the square of the 
circle, thus : 2x2=4. 

So 4 is the square, or 4 is the area of the circle ; and adding 
the area of the dodecagon, equal 108, to 4, makes 112, the area 
of the round table ; but to measure a circle, as people generally 
do, the same as square measure, from an angle of 90 degrees, 
it would make 6 instead of 4 for the square of the circle. 

But if people understood how to work trigonometry on a 
circle, it would prove but 4, or by the rule of three ; if 90 
give 6, what will 60 give? Answer, 4. 

To find the measure of a true oval : Suppose 
the conjugate be 12 inches, and the transverse , 
16; multiply 16 by 12; then divide the prod- 
uct by 9, and f- will be the area. 

To find the length to go round the oval : Take the measure 
of the conjugate, equal to 12, and multiply it by itself, thus : 
12 X 12 = 144. Then multiply the transverse, equal to 16, 
by itself, and the product is 256 ; subtract the product of the 
conjugate, equal 144, from it, which leaves 112; extract the 
square root of 112, equal 10.583 ; divide the root by 2, which 
gives 5.292. This is the erect leg. Then get the circumfer- 
ence of a circle 12 inches in diameter, which is 38 inches, and 
divide it by 4, which gives 9.5 for the base ; now square the 
two legs, and the square root of the sum will be one fourth of 
the circumference, which if multiplied by 4 gives the whole 
circumference. 

To show that the measure to go round a circle is just ■£? part 
longer than a straight rod which measures the same area on 
the oblong square: Take 12 for a diameter; 3 times 12 are 
36, for the six sides of the hexagon, commonly, called the cir- 
cumference. The radius is 6 ; so I multiply 36 by one half 




142 THE MEASURE OF THE CIRCLE. 

the radius, equal to 3, or one half of 36, equal to 18, by 6 ; 
either way gives the area of the oblong square, 18 by 6, equal 
to 108, which is just the measure of the dodecagon, or six 
oblong squares, 6 by 3, as shown in the plate. 

Now for the measure of the circle. Find the circumference 
by the ratio, 9.5, which is just 38, gaining T V, equal to square 
measure of 90 degrees. For example: Take a straight strip 
38 inches long, for circumference of the longest ring, gaining 
2 inches, which is T \. Then set a bevel of 90 degrees ; divide 
the 2 gained into 12 equal parts, making 2 inches, square meas- 
ure, from 90 degrees. Then alter the bevel to 60 degrees, and 
mark it off just at the same distance as you did at 90 degrees, 
and you will find you lose just one third of your measure. 
Now, 60 degrees make the only equal triangle that can be 
made, and is the right angle to measure a circle, as one of 90 
degrees is right for a square. Therefore, if 90 gives T V, what 
will 60 give ? Answer, ^ T . 

So 108 gains T ^ on the square of 90 degrees, equal to 6 
inches, which gives 114, area; but at 60 degrees it gains only 
sV> equal to 4 inches ; so if 90 gives 6, 60 gives 4 ; and 4 
added to 108 makes 112, the true area. Now, 112 makes an 
oblong square just 18|- by 6 ; so you see it takes -fa more to 
measure a circle than it does to measure the two sides of the 
oblong square, for the oblong square is straight measure. 

Proof: As the circle gains 2, and the square of 2 is 4, which 
is the square of the circle, so the 108 gains 2 y, equal to 4, 
making 112, area. So you see the area proves the circle, and 
the circle proves the area to be perfect measure. 

On a former measurement of the circle that has come under 
my observation, the circle being 12 inches in diameter, they 
called the circumference 37xV Now, suppose this to be the 
right measure ; then three times the diameter, 36, would make 
an oblong square 18 by 6, equal 108 inches. Take 3 times 
12, equal to 36, from 37-^, which leaves 1 inch and T 7 -, which 
is what it gains by being brought to a circle. So I square it, 
thus : — 



THE MEASURE OF THE CIRCLE. 143 



17 
17 



289 



So you may see that 2 inches and 89 hundredths is the square 
of the circle ; I then add the square of the circle to the oblong 
square, thus : — 

108 
2.89 



110.89, area. 

But if you add the angle that it gains, and work it as 90 
degrees, equal square measure, it gains 5 T V, "which added to 
108 makes 113 T V, area. By this you add the triangle of square 
measure, instead of the square of the circle ; but if they had 
brought 38 circumference, its area would be 114, by their way 
of working it. 

The great circle of this earth is what measure is derived 
from. Long measure, square measure, cubic measure, dry 
measure, and all measure, is to be derived from an equal pro- 
portion of this great circle, of and in which we exist ; and 
unless that equal proportion is found, no measure is or can be 
correct ; and that equal proportion must be found by a perfect 
measure of the great circle, Now, as an inch was calculated 
to be derived from an equal proportion of the great circle, and 
this derivation was from the proportion of as 7 to 22, the inch 
is imperfect, unless the proportion as 7 to 22 is right ; and if 
the inch is imperfect, the foot, the yard, the rod, the mile, the 
league, and all other space and distance that is calculated to be 
derived from proportion of equality, must be also imperfect. 

The proportion of equality that is derived from as 7 to 22 is 
imperfect, for the reason that the circle will not prove the area, 
nor the area prove the circle ; neither will the circle prove 
the square, nor the square prove 'the circle ; nor will the square 
prove the area. Now, 1728 cubic inches of water are calcu- 



144 THE MEASURE OF THE CIRCLE. 

lated to be one foot ; and according to specific gravity, the foot 
of water weighs 62s- pounds. From this pounds and ounces 
are calculated ; an ounce is a certain number of inches of this 
water ; and the correctness of pounds and ounces must be 
derived from an equal proportion of the circle, the same as 
measurement. 

This proportion, as 7 to 22, is so imperfect in the operation 
of mathematics, that in astronomical calculations it makes math- 
ematics useless. Thus it has been considered, since the present 
operation of mathematics has been known ; and instead of using 
mathematics in astronomical calculations, observatories have 
been erected, to observe distance, and space, and equality, for 
purposes to benefit mankind, which mathematics would not per- 
form with correctness, in its present state of imperfection, which 
imperfection has been for the want of a perfect quadrature of 
the circle. These observations are attended with great ex- 
pense, when compared with mathematics, with a perfect meas- 
ure of the circle. 

The measure of the circle shows that if you travel on a 
radius of one hundred miles you are 5J- miles astray, for the 
want of perfect measure. Also, in the traverse of the earth 
round the sun, the calculations have been 1,500,000 miles in 
24 hours ; whereas correct measure makes it 1,700,000. Also, 
it shows that no level can be found on the earth, for the want 
of a perfect measure. It shows that the survey between the 
two oceans, at the Isthmus of Darien, was very imperfect, for 
the want of the perfect measure of the circle. 

I can see no necessity for altering the names applied in long 
measure, square measure, dry measure, or in any measure, 
except the inch. A mathematical inch is the thirty-eighth part 
of a circle 12 inches in diameter. When the inch is made per- 
fect, the foot, the yard, the rod, the degree, the league, &c, &c, 
are all correct. 

The fundamental principles and bases upon which mathemat- 
ics is built have slept in oblivion since the world began. Space 
and distance, with equality with the great circle of the earth, 



THE MEASURE OF THE CIRCLE. 145 

have never been known by any mortal man. That equality of 
the great circle of this earth, that rectifies all weights and 
measures, which is a great cause of quietude, peace, and hap- 
piness with mankind, has slept in oblivion, incomprehensible to 
man. It has been the strife and anxiety of all the most learned 
and the lovers of science, since the commencement of the 
world; large sums of money, and much time and attention, 
have been made use of by all nations to effect a perfect measure 
of the circle. 

John Q. Adams's report of 1821, on coins, weights, and 
measures, will give a perfect history of the great exertions, 
strife, and anxiety with all nations for this measure. Mr. 
Adams says it has been the object of all statute making to guard 
against fraud and oppression, in consequence of the imperfec- 
tion of weights and measures. 

Space and distance can now be equalized by some measure 
of invariable length as a standard, which has been hitherto 
sought. Every man, woman, and child can now be equal in 
weight and measure, which is joy to their souls and peace to 
their minds. 

The prevailing opinion in the United States is, that we have 
a standard of weights and measures ; that no unrighteousness 
is done in mete or yard ; a just balance, a just weight, a just 
ephah, a just hin, is measured and weighed to all. 

We have in the United States some 48 custom houses. One 
house uses for a bushel 1925 inches, another, 2063, another, 
2216, another, 2203, another, 2158, another, 2358; and in 
this way does it vary, making the average variation in all 75 
inches in a bushel ; and so it is all over the world. 

This measure is now comprehended ; equality, distance, and 
space can now be measured to perfection ; all weights and meas- 
ures, gold and silver coin, can now be determined, to the satis- 
faction of every man, woman, and child. That law, not by 
human legislation, but by God, on Mount Sinai, can now be 
fulfilled ; a just balance, a just weight, a just ephah, a just hin, 
can now be made to all the world. 
13 



146 THE MEASURE OF THE CIRCLE. 

Having the circumference, to find the diameter : Divide the 
circumference by 19, and multiply the product by 6, which 
gives the diameter. 

The hexagon, with the number 6, is what measures the 
circle. 

It is the fourth power, or biquadrate, which makes up the 
measure of the circle. 

It is what is contained beyond the dodecagon ; it is -fa in the 
circle, and ^V in its area. 

The circle is perfectly synonymous with the square. 

The square of the circle may be considered by some to be 
the square that 12 diameter in a circle would make, which 
would be 9£ inches. 

I consider the square of the circle to be the square of the 
square, which is the square of 2, thus : 2x2=4. 

Square measure is from an angle of 90 degrees; circular 
measure is from an angle of 60 degrees. 

Circular measure gains over all other measure ; once in diam- 
eter is four times in area. 

To measure a globe or sphere : Suppose 12 is the diameter; 
first find the circumference, which is 38 inches ; then multiply 
it by the diameter, 12 ; 12 X 38 = 456 ; divide 456 by 19, 
which gives 24 ; add the 24 to 456, equal 480 inches, the sur- 
face of a globe 12 inches in diameter. The surface of any 
globe may be found in the same way. 

To find the solid contents of the same : First find the square 
of the diameter, thus : 12 X 12 = 144 ; multiply 144 by 12, 
which gives 1748, cube ; divide the cube by 12, which gives 
144 ; multiply 144 by 7, which gives T T ? , or 1008, the solid 
contents of a globe 12 inches in diameter. Any globe may be 
measured in this way. 

For example : Suppose a globe 6 inches in diameter : — 



THE MEASURE OF THE CIRCLE. 147 

19, circumference. 
6 

19)114(6 
114 



Add the 6 to 114 
6 

120, surface. 

Surface twice in diameter gives 4 times surface. Globe twice 
in diameter gives 8 times as much solid contents. 

6, solid contents. 
6 

86 
6 

12 ) 216, cube. 

18 

7 

126, solid contents of a globe 6 
inches in diameter. 

To find any one side of a right angled triangle, having the 
other two sides : In every right angled triangle the square of 
the hypotenuse is equal to both the squares of the two legs. 
Therefore, to find the hypotenuse add the square of the two 
legs together, and extract the square root of the sum. And to 
find the leg, subtract the square of the other leg from the 
square of the hypotenuse, and the square root of the difference 
is the leg required. 

Example 1 • What is the hypotenuse of a right angled tri- 
angle whose base is 56, and perpendicular 33 ? 



148 THE MEASURE OF THE CIRCLE. 

56 33 

56 33 

336 99 

280 99 

3136 1089 

1089 

4225 ( 65 feet, answer. 

Example 2. What is the perpendicular of a right angled tri- 
angle whose base is 40 yards, and hypotenuse 50 yards ? 

40 50 

40 ^ 50 

1600 2500 

' 1600 

900 yards, answer. 

Example 3. "What is the height of a scaling ladder to reach 
the top of a wall 28 feet in height, and across a ditch 45 feet 
in breadth? Answer, 50 feet. 

Questions of this nature are resolved by the foregoing prob- 
lem ; the height of the ladder being considered as the hypote- 
nuse of a right angled triangle, and the height of the wall and 
breadth of the ditch as the two other legs of the triangle. 

CIRCULAR ROOT. 

To find the hypotenuse on a circle, having the diameter of 
any circle given : First find the circumference, and take any 
part of the circle, or the whole of the circumference, as needs 
may require, and call it the base, and square the same ; then 
take the erect leg, and square it also, and add the two sums 
together, as in extracting the square root ; and the square root 
of the sum will be the circular hypotenuse. 



THE MEASURE OF THE CIRCLE. 149 

By this rule you may find the length of a circular stair rail, 
or any thing that may be wanting of that circular form, just 
the same as the square root. 

Example : Suppose a circular flight of stairs 12 feet high 
and 6 feet in diameter, equal 19 feet base; what is the length 
of the hand rail ? 

19 12 

19 12 

171 144, square of leg. 

19 

361, square of base. 
144 

V 505 ( 22.472 
4 

42 ) 105 
84 



444 ) 2100 
1776 



4487 ) 32400 
31409 



44942)99100 

89884 





Answer : the stair rail is 22 feet and 472 thousandths of a foot 
long. 

Space and distance, with equality of the great circle of this 
earth, have never been found; consequently, no man knows 
what an inch is, a foot, a yard, a rod, a league, a mile, a de- 
gree, a pound, an ounce, a glass, a gill, a half pint, a pint, a 
quart, a gallon, a peck, a half bushel, a bushel. 
13* 



150 THE MEASURE OF THE CIRCLE. 

These measures and weights enter the concerns, privately, 
individually, domestically, of each and every individual upon 
the earth, young and old, rich and poor, great and small, from 
the king to the beggar. Equality and space are the funda- 
mental principles and bases of moral justification of right and 
wrong ; without these no man knows whether he is rightly used 
or wrongfully abused. They are the mediators of contentment 
and justice, and teach mankind to do right. They enable men 
to have confidence and quietude in all their transactions with 
each other. They are the mediators between the honest man 
and the knave, and steps to salvation that make glad the heart 
of man. 

It is the supposition and belief of the majority of mankind 
that we have a standard of weights and measures throughout 
the world ; but this is erroneous ; there is no standard of 
weights and measures, because such a standard would be uni- 
form, and such a thing as uniformity of weights and measures 
is not known in this world. For example, England may have 
what she calls a standard, but this standard will vary in every 
state and kingdom, town and county ; no two are alike, and 
there is no one that is right. How can it be possible to make 
a correct standard of weights and measures, when that correct- 
ness proceeds from a perfect equality of the circle of the earth, 
and that equality has never been known by any living man or 
mortal since the world began ? That equality has been incom- 
prehensible to man; therefore justice and honor to all men, 
as regards weights and measures, has been wavering to the four 
cardinal points. If a man gets a pint, quart, or gallon, it is 
well, and if not, all the same ; for who is to say whether it is 
mete or measure ? As figures are inexpressible to man, he can- 
not determine its equality. To obtain this measure has been 
the continual exertion of England, France, and Spain, for hun- 
dreds and thousands of years.** At one period France kept 
twenty or thirty of her most scientific and learned men for seven 
years, endeavoring to arrive at some determined equality of 



THE MEASURE OF THE CIRCLE. 151 

measure. The ambition of these men was so much aroused 
that they sailed round the world. 

Thomas Jefferson says, the first object that presented itself 
was the discovery of some measure of invariable length, as a 
standard. There exists not in nature, he says, as far as has 
been hitherto observed, a single subject, or species of subject, 
accessible to man, that presents a uniform dimension. , 



OBSERVATIONS. 

I have carefully collected these ideas, thinking they might 
be of some use to my fellow-men ; and if 1 have accomplished 
my desire to the satisfaction of my readers, my prayers are 
answered ; and if not, it cannot be imputed to the want of an 
honest intention. 

The brute creation may perhaps enjoy the faculty of behold- 
ing visible objects with a more penetrating eye than ourselves ; 
but spiritual objects are as far out of their sight as though they 
had no being. Nearest to the brute creation are those men 
who suffer themselves to be so far governed by external objects 
as to believe nothing but what they can comprehend with their 
shallow and imaginary understandings, and who let such 
expressions as these fall from their lips, in floods of abun- 
dance : — "O, it is impossible — it cannot be — no reasonable 
man can think so ! " as was said to me, when endeavoring to 
convince them of my discovery. I was told that no rational 
mathematician would ever make the attempt. Proud, vain, 
and foolish man ! how unwise art thou, for the want of that 
good reasoning which Jesus Christ dictated to the world ! Let 
us all gain wisdom by moderate, cool, thoughtful closet prayer ; 
consult in private with your heavenly Father, for he is your 
God, and not man. Let all your meditations be with him, and 
not with man. Your conscience is the mediator between your- 
self and God. This putting trust in and worshipping men and 
idols may be imputed to the representation made use of regard- 



152 THE MEASURE OF THE CIRCLE. 

ing beasts, as confined to earthly objects, and not to our heavenly 
Father, who seeth all things on earth as well as in heaven. 

If I have done myself a credit, I hope it may be imputed as 
an honor to man; I wish not to boast, as it is not of me. 
Friendship, harmony, love, universal kindness to all men, and a 
fervent desire for peace and contentment after this life, are the 
heart throbbings and conscientious feelings of your humble 
servant. 



TESTIMONIALS 



I have travelled with Mr. John Davis, in England and the 
United States, for six years; we have visited all the most 
learned mathematicians that we could hear of, but have never 
found one that attempted a disapproval of his work. I have 
examined, to the best of my ability, the works of writers on 
mathematics since the time of Euclid ; and without a doubt, 
according to mathematics, he has solved the wonderful problem, 
surprising as it may appear to many. 

Weights and measures can now be made perfect, and to the 
satisfaction of every man, woman, and child. Space and dis- 
tance, which have never been known, are now found by John 
Davis. Sabin Smith, 

Attorney and Agent for John Davis. 



Nothing is more recreating or interesting to a mathematical 
mind than John Davis's measure of the circle. It has been the 
study and strife of our learned and scientific men since the time 
of Archimedes ; and it is a manifestation of supremacy that 
this measure is perfected. Its value pen or tongue cannot de- 
scribe. I anticipate great improvements from this measure in 
the next fifty years, to the pride, happiness, and benefit of man- 
kind. No doubts can rest with me as to the perfection of this 
measure. Thomas Rossiter. 

(153) 



154 TESTIMONIALS. 

I have had an opportunity for two years to be with Mr. 
Davis, and during this time I have given great attention to his 
work — the measure of the circle. It has been my business, 
since my manhood, and always was and still is my pride, to 
practise in mathematical operations, such as surveying, &c. I 
have' had transactions with many able mathematicians, but have 
never met with such a man as John Davis. He has a mind 
which exceeds, in the comprehension of mathematics, all others 
with which I am acquainted. His measure of the circle is a 
wonderful discovery, and no doubts can be entertained of its 
perfection, when mathematically demonstrated. 

Asahel Buck. 



I have wrought as a mechanic, at the tin ware business, for 
fifteen years. I became acquainted with Mr. Davis in New 
York city, while working at my business. Mr. Davis and Mr. 
Smith taught me the measure of the circle, so that I could find 
the circumference of a circle, and its area. I have made use 
of his measure in my work, and find it perfect. 

Charles Wilkinson. 



New York, July 17, 1850. 
I have taken into consideration the measurement of the 
circle, and have examined it to the best of my ability, giving it 
much time and attention ; and do say that, to the best of my 
belief, a complete measurement has been effected by Mr. John 
Davis. M. F. Greenlaw. 



I have examined the measure of the circle by Mr. John 
Davis, and find it, in my opinion, a most complete, scientific, 
mathematical measure of the circle. 

Thomas Guile, 
Professor of Mathematics. 



TESTIMONIALS. 155 

I have wrought as a mechanic for twenty years, and in some 
of my mechanical operations I have found it very difficult to 
match my work from the proportion of as 7 to 22, and by 
experimental operations I came to the measure of three times 
the diameter, and one sixth, and from this I have found no dif- 
ficulty in matching my work ; and when Mr. Davis told me 
that three and one sixth times the diameter was his proportion, 
I was satisfied that his measure was correct. 

A Mechanic of Paterson, N. J. 



I have examined the measure of the circle by John Davis, 
and beyond all doubt it is perfect measure. 

Henry R. Savory, 
Civil and Military Engineer. 



"THE CIRCLE OF THE FAIR." 

BY THOMAS ROSSITER. 

We all have had a tangle 

In the " circle of the fair," 
And stricken by a charm 

In our bosom — funny, queer : — 
But who can straightly tell us, 

Independent of a doubt, 
Once in the blessed circle, 

How the deuce they did get out ? 

Remembering the spider's 

Habitation for a fly, 
We're fearful to encounter, 

Yet seldom fail to try ; 
For love is more enticing 

Than the pearls of a sham, 
And seldom fails to better 

Or subdue the inward man. 

Once, in the Eagle valley, 

By the Hudson's gentle tide, 
I saw a faithful shepherd 

Press a maiden to his side ) 
She was telling how she loved him, 

And how heightened was her bliss ; 
Though the flock strayed on the wolf-path, 

He was kissing her for this. 

Now, if we are unlucky 

In the " circle of the fair," 
We need but welcome Davis, 

He can for us make it square ; 
So drink a health to Davis, 

While the goddess of our theme 
Shall raise the wreath of Flora, 

As an emblem of his fame. 

.(156) 




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